Existence of solutions for some nonlinear elliptic problems · iv Index 2.9 The Palais-Smale ... This thesis deals with the study of some nonlinear elliptic problems coming from Mathematical

UNIVERSITA DEGLI STUDI DI BARI

Dottorato di Ricerca in Matematica

XIX Ciclo – A.A. 2005–2006

Settore Scientifico-Disciplinare:

MAT/05 – Analisi Matematica

Tesi di Dottorato

Existence of solutions for somenonlinear elliptic problems

Candidata:

Sara BARILE

Supervisori della tesi:

Prof. David ARCOYA

Prof. Silvia CINGOLANI

Coordinatore del Dottorato di Ricerca:

Prof. Silvia ROMANELLI

Contents

Introduction 1

Acknowledgements 5

I Magnetic Schrodinger equations 7

1 Magnetic Schrodinger equations with critical exponents 9

1.1 A review on some perturbation methods in Critical Point Theory 9

1.2 A degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Single–peaks for a magnetic Schrodinger equation with criticalgrowth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 The limiting problem . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 The variational framework . . . . . . . . . . . . . . . . . . . . . 25

1.6 Asymptotic study of Γ . . . . . . . . . . . . . . . . . . . . . . . 28

1.7 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . 45

1.8 Regularity results . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.9 The regularity of solutions involving the operator ∆A. . . . . . 47

2 Magnetic Schrodinger equations with singular electric poten-tial 57

2.1 A review on Ljusternik-Schnirelman theory . . . . . . . . . . . 57

2.2 Minimax theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3 A multiplicity result . . . . . . . . . . . . . . . . . . . . . . . . 61

2.4 A multiplicity result for singular NLS equations with magneticpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.5 Magnetic fields: the space HA . . . . . . . . . . . . . . . . . . . 66

2.6 The variational framework . . . . . . . . . . . . . . . . . . . . . 67

2.7 The limit energy level . . . . . . . . . . . . . . . . . . . . . . . 71

2.8 The barycentre map . . . . . . . . . . . . . . . . . . . . . . . . 74

iv Index

2.9 The Palais-Smale condition . . . . . . . . . . . . . . . . . . . . 832.10 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . 91

II Blow-up solutions for semilinear problems 93

3 Singular quasilinear equations with natural growth and semi-linear problems with blow-up at the boundary 953.1 Electrohydrodynamics. The equilibrium of a charged gas in a

container . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . 973.3 Existence, uniqueness and monotonicity of the solution . . . . . 983.4 Bounds on the solution . . . . . . . . . . . . . . . . . . . . . . . 993.5 Example: the ideal gas . . . . . . . . . . . . . . . . . . . . . . . 1003.6 A quasilinear approach for semilinear problems with blow-up at

the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.7 The approximated quasilinear problems . . . . . . . . . . . . . 1053.8 Positive solutions for the quasilinear problem . . . . . . . . . . 1113.9 Semilinear problems with blow-up at the boundary . . . . . . . 114

Bibliography 117

Introduction

This thesis deals with the study of some nonlinear elliptic problems comingfrom Mathematical Physics. It consists of two main parts.In Part I we study existence of standing waves solutions ψ(x, t) = e−iEt~

−1u(x),

E ∈ R, t ∈ R, u : RN → C, to the time-dependent nonlinear Schrodinger (NLS,

to be short) equation with electromagnetic potentials

i~∂ψ

∂t=

(~

i∇−A(x)

)2

ψ + U(x)ψ − f(|ψ|2)ψ, (0.0.1)

where ~ is the Planck constant, L~

A,U :=(

~

i∇−A(x))2

+ U(x) denotes aSchrodinger operator with a real-valued magnetic vector potential A and areal-valued electric scalar potential U , namely

L~

A,U := −~2∆ −

2~

iA · ∇ + |A|2 −

~

idivA+ U

and f(|ψ|2)ψ is a nonlinear term. For the spectral properties of L~

A,U we referto Avron, Herbst, Simon [15] and Helffer [63, 64].

By a mathematical point of view, the search of standing waves ψ to (0.0.1)leads to seek for solutions u : R

N → C for the complex semilinear ellipticequation in R

N :

(~

i∇−A(x)

)2

u+ V (x)u− f(|u|2)u = 0, (0.0.2)

where V (x) = U(x)−E. Such a problem is variational in nature and solutionsto (0.0.2) correspond to critical points of suitable functionals in Sobolev spaces.Starting from the well-known pioneering results due to Floer and Weinstein[57] and Rabinowitz [89], a large amount of international researches has beendevoted, in the last twenty years, to the study of NLS equations in the case A =0 (no magnetic) by different approaches [4, 39, 40, 51, 52, 62, 74, 82, 105, 106].

2 Introduction

The first existence result, by means of variational methods, for magnetic NLSequations (A 6= 0) with V (x) = 1 and a nonlinear term f having a subcrit-ical growth (in the sense of Sobolev immersions) is due to M. Esteban andP.L. Lions [55]. More recently, existence and multiplicity results have beenobtained by several authors; among them, Kurata [70], Cingolani and Secchi[44, 45, 47], Chabrowski and Szulkin [36], Bartsch, Dancer and Peng [19] andCao and Tang [35].In Chapter 1, we present existence results for a class of magnetic Schrodingerequations in R

N with critical growth by means of perturbation techniques,in Critical Point Theory, introduced by Ambrosetti, Badiale in [6], by Am-brosetti, Badiale, Cingolani in [4] for no magnetic NLS equations, and byAmbrosetti, Malchiodi in [8] for the Yamabe problem. Precisely, we look forsolutions u : R

N → C of the semilinear elliptic equation

(~

i∇− εA(x)

)2

u+ εαV (x)u = |u|2∗−2u in R

N , (0.0.3)

where N > 4, ε is a positive small parameter, 2∗ := 2N/(N − 2), α ∈ [1, 2],A : R

N → RN is a bounded magnetic potential and V : R

N → R is a boundedelectric potential. Firstly we prove that the unperturbed problem

−∆u = |u|2∗−2u in R

N ,

(i.e., (0.0.3) with ε = 0) has a finite-dimensional and non-degenerate manifoldof critical points

Z =eiσzµ,ξ : σ ∈ S1, µ > 0, ξ ∈ R

N∼= S1 × (0,+∞) × R

N

where

zµ,ξ(x) = κNµ(N

2−1)

(µ2 + |x− ξ|2)N−2

2

, κN = (N(N − 2))N−2

4 .

Next, performing a finite-dimensional reduction for (0.0.3), we construct afinite-dimensional natural constraint Zε for fε. At this point, we find that(stable) critical points of an auxiliar map -called the Melnikov function- Γ :Z → R correspond to points on Z from which solutions to (0.0.3) bifurcatefor ε small. If V changes its sign, we find an existence result for at least twosolutions. This result has been obtained in a joint work with S. Cingolani andS. Secchi [18].Finally we show that certain regularity properties of solutions for a rather

Introduction 3

general class of Schrodinger equations with electromagnetic field, establishedin the paper [36] by Chabrowski and Szulkin, hold true for the branch ofsolutions to (0.0.3).In Chapter 2, we discuss a multiplicity result of stationary solutions u : R

N →C, in the semiclassical limit (~ → 0), of the singular nonlinear Schrodingerequation with an external electromagnetic field

(~

i∇−A(x)

)2

u+ (V (x) − ε(~)W (x))u = |u|p−2u in RN , (0.0.4)

where N ≥ 2, 2 < p < 2∗ ( where 2∗ = ∞ if N = 2 and 2∗ = 2N/(N − 2)if N ≥ 3 ), A ∈ C1(RN ,RN ) is a magnetic potential, V ∈ C(RN ,R) is apositive electric potential, possibly unbounded from above, ε = ε(~) is a non-negative function depending on ~ > 0, W : R

N → R is a positive measur-able function which may be singular, so that the perturbing electric potentialV (x)− ε(~)W (x) may be unbounded below. Our result covers the meaningfulphysical cases of W (x) = 1

|x| , W (x) = 1|x|2 and constant magnetic fields having

source in potentials A(x) with polynomial growths. By using topological ar-guments introduced by Coron [48] and Benci-Cerami [20, 21, 22], we find thatthe number of solutions to (0.0.4) is bounded by some number which dependson the category of the set of the global minima of V , for ~ sufficiently small.Such result is contained in a recent paper by S. Barile [17].

In Part II we study existence results to singular quasilinear elliptic problemswith “natural” growth and semilinear elliptic problems with blow-up at theboundary after we have established the interplay between such classes of equa-tions.In Chapter 3, we start recalling a first existence result for blow-up solutionsdue to Keller [68] and Osserman [83] and how Keller has applied such resultin [67] to electrohydrodinamics, namely to the problem of the equilibrium ofa charged gas in a conducting container.Then, we discuss a recent result proving the existence of positive solutions toquasilinear elliptic problems having a gradient term with quadratic growth ofthe type

−∆w + g(x,w)|∇w|2 = a(x), x ∈ Ω,

w ∈ H10 (Ω),

(0.0.5)

where Ω is a bounded domain in RN (N ≥ 3), a ∈ Lq(Ω) for some q > N/2

and g : Ω× (0,+∞) → R is a continuous function in (0,+∞) which may have

4 Introduction

a singularity at zero and may change of sign. Our purpose is to improve somerecent existence results (see [12, 13]) to (0.0.5) where the case of a positivenonlinearity g(x, s) in Ω× (0,+∞) with a singularity in s = 0 is studied. Thesimplest model which we refer to is represented by the equation

−∆w +1

w|∇w|2 = a(x), x ∈ Ω.

It is worth while to remark that the study of a not necessarily positive andsingular nonlinearity g is completely new even in the framework of quasilinearelliptic equations with “natural growth”. Indeed, our result constitutes alsoan improvement of the earlier studies of (0.0.5) by Bensoussan, Boccardo,Murat and Puel for a nonsingular Caratheodory (thus continuous in the secondvariable) g : Ω × R → R verifying the sign condition

sg(x, s) ≥ 0, ∀s ∈ R, a.e. x ∈ Ω,

(see [23, 31, 32]) and for a general Caratheodory term g(x,w,∇w) which sat-isfies a more general “one-side condition” that, in the case of a pure quadraticterm, is not more than the above sign condition (see [30]).The study of this class of quasilinear equations is even motivated by the searchof positive solutions for semilinear elliptic problems of the type

∆u = a(x)f(u), x ∈ Ω,

u = ∞, x ∈ ∂Ω,(0.0.6)

where f ∈ C1[0,+∞) is a strictly positive function on (0,+∞), arising fromthe study of Riemannian surfaces of constant negative curvature and the sub-sonic motion of a gas. Such solutions, usually known as large solutions, havebeen studied by many authors like Bandle and Marcus, Keller, Lair, Osserman,Tao and Zhang [16, 68, 71, 83, 102].

Differently from these works, where the sub and super-solution method isessentially used, we develop a new approach to solve (0.0.6). Indeed, by asuitable nonlinear change of variable, we establish an equivalence between thesearch of solutions to this problem and the existence results to the quasilinearproblem (0.0.5) previously obtained. It is interesting to underline that theimprovement obtained to handle a not necessarily nonnegative g in (0.0.5)corresponds for (0.0.6) to overcome the requirement in [16, 68, 71, 83] of somemonotony properties on f (for example, f nondecreasing). Such results arecontained in a joint work with D. Arcoya and P.J. Martınez Aparicio [11].

Introduction 5

Acknowledgements.I am greatly indebted to Prof. Silvia Cingolani for teaching me the subject,for the stimulating discussions, for the continuous support and encouragementthat she gave me during the development of my research program.I gratefully acknowledge Prof. David Arcoya for the fruitful collaboration,the support and the kind hospitality during my stay in Granada (Spain). Iexpress my special gratitude to Prof. Enrico Jannelli for the constant interestand Prof. Dino Fortunato for the support in my research activity. I wishalso to thank Dr. Simone Secchi, Dr. Pedro J. Martınez Aparicio and all theDepartment of Mathematics in Granada for the nice collaborations.

Part I

Magnetic Schrodinger

equations

Chapter 1


equations with critical

exponents

1.1 A review on some perturbation methods in Crit-

ical Point Theory

In this Section we give the main ideas and results of a variational methodto study critical points of perturbed functionals and to deal with problemswith lack of compactness. The method has been developed by Ambrosettiand Badiale in [5, 6], extending some previous results contained in [2, 4]. Itallows us to treat problems in which a “small parameter” appears and permitsto find existence and multiplicity results even in some situations where theConcentration-Compactness methods of P.L. Lions [76, 77] fails or requiresheavy calculations. The key idea is to perform a suitable finite dimensionalreduction and to find the solutions of the “perturbed” problem near the solu-tions of the “unperturbed” one.We deal with a family of C2 functionals fε, defined on a Hilbert space E,parametrized by ε ≥ 0. We suppose that the “unperturbed” functional f0 hasa whole manifold of critical points. Under suitable assumptions, it is possibleto prove the existence of critical points of each fε, for all ε sufficiently small.For the sake of completeness, we present below this technique in some details(see Ambrosetti, Malchiodi [10]).We want to find critical points of functionals of the form

10 Chapter 1. Magnetic Schrodinger equations with critical exponents

fε(u) = f0(u) + εG(u)

where f0 ∈ C2(E,R) plays the role of the unperturbed functional and G ∈C2(E,R) is the perturbation.We will always suppose that

(CM) there exists a d-dimensional smooth, say C2, manifold Z, 0 < d =dimZ < ∞, such that each z ∈ Z is a critical point of f0. The set Zwill be called a critical manifold of f0.

Remark 1.1.1. The presence of a critical manifold is usually due to thefact that the unperturbed functional f0 is invariant under the action of asymmetry group: for example, in the case discussed in the next Sectionsit will be invariant under translations and scale changes.Since fε does not satisfy, in general, the Palais-Smale condition, Z willalways be not compact. Then, we will investigate in which circumstancesthe perturbation G makes it possible to recover the compactness andallows us to find critical points of fε.

Let TzZ denote the tangent space to Z at z. If Z is a critical manifoldthen for every z ∈ Z one has that f ′0(z) = 0. Differentiating this identity,we get

(f ′′0 (z)[v]|φ) = 0, ∀v ∈ TzZ, ∀φ ∈ E,

and this shows that every v ∈ TzZ is a solution of the linearized equationf ′′0 (z)[v] = 0, namely that v ∈ ker[f ′′0 (z)]: TzZ ⊆ ker[f ′′0 (z)]. In particu-lar, f ′′0 (z) has a non trivial Kernel (whose dimension is at least d) andhence all the z ∈ Z are degenerate critical points of f0. We shall requirethat this degeneracy is minimal. Precisely we will suppose that

(ND) TzZ = ker[f ′′0 (z)], ∀z ∈ Z.

Remark 1.1.2. Assumption (ND) is a sort of nondegeneracy condition.Actually, if instead of a manifold, we consider an isolated critical pointu0, i.e. Z is an isolated point u0, the condition (ND) corresponds torequire that f ′′0 (u0) is invertible, namely that u0 is non-degenerate criticalpoint of f0. Obviously, in such a case, a straight application of theImplicit Function Theorem allows us to find, for |ε| small, a solution of

f ′ε(u) = 0, ∀u ∈ E.

1.1. Some perturbation methods in Critical Point Theory 11

Unlike, dealing with a critical manifold, proving that Z satisfies (ND)is equivalent to show that ker[f ′′0 (z)] ⊆ TzZ, namely that every solutionof the linearized equation f ′′0 (z)[v] = 0 belongs to TzZ.

In addition to (ND) we will assume that

(Fr) for all z ∈ Z, f ′′0 (z) is an index 0 Fredholm map.1

Definition 1.1.3. A critical manifold Z will be called non degenerate, NDin short, if (ND) and (Fr) hold.

If the preceding assumptions hold true, one can use the Implicit FunctionTheorem to produce a good local deformation of Z in the normal direction.More precisely, we have the following lemma.

Lemma 1.1.4. Suppose f0 satisfies (CM)− (ND)− (Fr). Then, given R > 0and BR = u ∈ E : ||u|| ≤ R, there exist ε0 and a smooth function w =w(ε, z) : M = (−ε0, ε0) × Z ∩BR → E such that

1. w(0, z) = 0 for all z ∈ Z ∩BR;

2. w(ε, z) is orthogonal to TzZ, for all (ε, z) ∈M , or equivalently w(ε, z) ∈(TzZ)⊥, and is of class C1 with respect to z ∈ Z ∩ BR. In particular,w(ε, z) → 0 as |ε| → 0, uniformly with respect to z ∈ Z ∩ BR, togetherwith its derivative with respect to z ∈ Z ∩BR, w′(ε, z);

3. ||w(ε, z)|| = O(ε) as ε→ 0, for all z ∈ Z ∩BR.

Letting Zε = z + w(ε, z) : (ε, z) ∈M, it turns out that Zε is locally diffeo-morphic to Z. Moreover we have the following result.

Lemma 1.1.5. Zε is a natural constraint for f ′ε, namely: if uε = z+w(ε, z) ∈Zε and f ′ε|Zε(uε) = 0, then f ′ε(uε) = 0 (any critical point of fε constrained onZε is a stationary point of fε).

Proof. Suppose that f ′ε|Zε(uε) = 0, then f ′ε(uε) = 0 is orthogonal to TuεZε.On the other side, f ′ε(uε) ∈ TzZ (by the definition of w(ε, z)) and TuεZε isnear to TzZ for ε small. Therefore f ′ε(uε) = 0.

1A linear map T ∈ L(E, E) is Fredholm if the kernel is finite-dimensional and the imageis closed and has finite codimension. The index of T is dim(ker[T ]) − codim(Im[T ])


We find that the behavior of fε on Zε is well approximated by the behaviorof the function Γ ≡ G|Z . In particular, critical points of Γ on Z give rise tocritical points of fε on Zε.Firstly, one shows that

fε(u) = f0(z) + εΓ(z) + o(ε),

for any u ∈ Zε and |ε| small enough. This in turn yields:

Theorem 1.1.6. Let f0 satisfy (CM) − (ND) − (Fr) and suppose that thereexists a critical point z ∈ Z of Γ such that one of the following conditionsholds:

(i) z is non-degenerate;

(ii) z is a local proper minimum or maximum;

(iii) z is isolated and the local topological degree of Γ′ at z, degloc(Γ′, 0) 6= 0.

Then for |ε| small, the functional fε has a critical point uε such that uε → zas ε→ 0.

1.2 A degenerate case

If G(z) ≡ 0 for all z ∈ Z, Lemma 1.1.4 is useless since the Melnikov functionΓ would vanish identically. Thus, we need to evaluate the further terms inthe expansion of fε and find a slightly implicit Melnikov function whose stablecritical points produce critical points of fε.For z ∈ Z, denote Lz the inverse of the restriction to (TzZ)⊥ of f ′′0 (z).

Lemma 1.2.1. If G(z) ≡ 0 for every z ∈ Z, then

w(ε, z) = −εLzG′(z) + o(ε) (1.2.1)

Proof. Let us recall that w satisfies 2. in Lemma 1.1.4 and Dfε(z + w) ∈TzZ, namely f ′′0 (z)[w] + εG′(z) + o(ε) ∈ TzZ. Moreover, G(z) ≡ 0 impliesG′(z)⊥TzZ. Therefore, w(ε, z) = −εLzG

′(z) + o(ε).

Let us now expand fε. One has:

fε(z + w(ε, z)) = f0(z + w(ε, z)) + εG(z + w(ε, z))

= f0(z) +1

2f ′′0 (z)[w,w] + εG(z) + εG′(z)[w] + o(ε2)

1.2. A degenerate case 13

Since G(z) ≡ 0, using Lemma 1.2.1 we infer

fε(z + w(ε, z)) = f0(z) −1

2ε2(LzG

′(z), G′(z))

+ o(ε2)

= f0(z) + ε2Γ(z) + o(ε2),

where

Γ(z) = −1

2

(LzG

′(z), G′(z)).

In the next Sections, as in the applications to the Yamabe problem on SN in[8] by Ambrosetti and Malchiodi, we will deal with a functional of the form

fε(u) = f0(u) + εG1(u) + ε2G2(u) + o(ε2), u ∈ E. (1.2.2)

It is always understood that E is a Hilbert space and f0, G1, G2 ∈ C2(E,R).In such a case the preceding arguments yield:

Lemma 1.2.2. Suppose that

G1(z) = 0, ∀z ∈ Z (1.2.3)

and let Γ : Z → R be defined by setting

Γ(z) = G2(z) −1

2

(LzG

′1(z), G

′1(z)

). (1.2.4)

Then we have

fε(z + w(ε, z)) = f0(z) + ε2Γ(z) + o(ε2). (1.2.5)

Proof. Since G1|Z ≡ 0, then G′1(z) ∈ (TzZ)⊥. So Lemma 1.2.1 holds. Then

one finds

fε(z + w(ε, z)) = f0(z + w(ε, z)) + εG1(z + w(ε, z)) + ε2G2(z + w(ε, z))

= f0(z) +1

2f ′′0 (z)[w,w] + εG1(z)

+ εG′1(z)[w] + ε2G2(z) + o(ε2).

Using (1.2.3) and (1.2.1) the claim follows.

We are now in position to state the abstract result that we will use to findthe critical points of fε.


Theorem 1.2.3. Assume that we are in the hypothesis of Lemma 1.1.4 andLemma 1.2.2 and that there exist a set A ⊆ Z with compact closure and z0 ∈ Asuch that

Γ(z0) < infz∈ ∂A

Γ(z) (respectively Γ(z0) > supz∈ ∂A

Γ(z)).

Then, for ε small enough, fε has at least a critical point uε ∈ Zε such that

f0(z) + ε2 infA

Γ + o(ε2) ≤ fε(uε) ≤ f0(z) + ε2 sup∂A

Γ + o(ε2)

(respectively f0(z) + ε2 inf∂A

Γ + o(ε2) ≤ fε(uε) ≤ f0(z) + ε2 supA

Γ + o(ε2)).

Furthermore, up to a subsequence, there exists z ∈ A such that uεn → z in Eas εn → 0.

1.3 Single–peaks for a magnetic Schrodinger equa-

tion with critical growth

This Section deals with some classes of elliptic equations which are perturba-tions of the time-dependent nonlinear Schrodinger equation

i~∂ψ

∂t= −~

2∆ψ − |ψ|2∗−2ψ (1.3.1)

under the effect of a magnetic field Bε and an electric field Eε whose sources aresmall in the L∞ sense. Precisely we will study the existence of wave functionsψ : R

N × R → C satisfying the nonlinear Schrodinger equation

i~∂ψ

∂t=

(~

i∇−Aε(x)

)2

ψ +Wε(x)ψ − |ψ|2∗−2ψ (1.3.2)

where Aε(x) and Wε(x) are respectively a magnetic potential and an elec-tric one, depending on a positive small parameter ε > 0. This model arisesin several branches of physics, e.g. in the description of the Bose–Einsteincondensates and in nonlinear optics (see [15, 28, 65, 80]). We assume thatAε(x) = ε A(x), Wε(x) = V0 + εαV (x), where A : R

N → RN and V0 ∈ R,

V : RN → R, α ∈ [1, 2].

On the right hand side of (1.3.2) the Schrodinger operator L~

Aε=(

~

i∇−Aε)2

denotes the formal scalar product of the operator ~

i∇−Aε by itself, i.e.

(~

i∇−Aε(x)

)2

ψ := −~2∆ψ −

2~

iAε · ∇ψ + |Aε|

2ψ −~

iψ divAε,

where i2 = −1, ~ the Planck constant. Actually, the magnetic field Bε isnothing but Bε = curlAε if N = 3; in general dimension, if A is seen as the1–form

A =N∑

j=1

Ajdxi,

then B should be thought of as a 2-form and

Bε = ε dA = ε∑

j<k

Bjk dxj ∧ dxk, where Bjk = ∂jAk − ∂kAj ,

represents the external magnetic field having source in εA (cf. [100]), whileEε = εα∇V (x) is the electric field. We notice that if we replace the magnetic


potential Aε by Aε(x) = Aε(x) + ∇ϕ(x) for some real-valued C2 function ϕ,

then Bε(x) = curlAε(x) = Bε(x) and

e−iϕ

[(1

i∇− Aε(x)

)2]eiϕ =

(1

i∇−Aε(x)

)2

,

so that the spectral properties of L~

Aεand L~

Aεare the same. The above

property is called the gauge invariance of the magnetic Schrodinger operatorand it is in accordance with the fact that the physically relevant quantity isthe magnetic field Bε and not its vector potential Aε. For fixed ~ > 0 thespectral theory of the operator has been studied in detail, particularly byAvron, Herbst, Simon [15] and Helffer [63, 64].The search of standing waves of the type ψε(t, x) = e−iV0~

−1tuε(x) leads to finda complex-valued solution u : R

N → C of the semilinear Schrodinger equation

(~

i∇− εA(x)

)2

u+ εαV (x)u = |u|2∗−2u in R

N . (1.3.3)

Dealing with the critical case, we mention the paper [14] by Arioli and Szulkin

where the existence of a solution is proved whenever 0 /∈ σ((

∇i −A

)2+ V

)

and the potentials A and V are assumed to be periodic, ε > 0 fixed. We alsocite the paper [36] by Chabrowski and Skulzin, dealing with entire solutions of(1.3.3) and the works [7, 43, 46], dealing with perturbed semilinear equationswith critical growth without magnetic potential A.In these Sections, V and A are not in general periodic potentials.

When the problem is nonmagnetic and static, i.e. A = 0, V = 0, and ~ = 1then problem (1.3.3) reduces to the equation

−∆u = |u|2∗−2u, u ∈ D1,2(RN ,C). (1.3.4)

In Section 1.4 we prove that the least-energy solutions to (1.3.4) are given bythe functions z = eiσzµ,ξ(x), where

zµ,ξ(x) = κNµ(N

2−1)

(µ2 + |x− ξ|2)N−2

2

, κN = (N(N − 2))N−2

4 (1.3.5)

and they correspond to the extremals of the Sobolev imbedding D1,2(RN ,C) ⊂L2∗(RN ,C) (cf. Lemma 1.4.1).

The perturbation of (1.3.4) due to the action of an external magneticpotential A leads us to seek for complex–valued solutions. In general, the

1.4. The limiting problem 17

lack of compactness due to the critical growth of the nonlinear term producesseveral difficulties in facing the problem by global variational methods. InSection 1.5 we will attack (1.3.3) by means of a perturbation method in CriticalPoint Theory borrowed from [6, 8, 10] and presented in the previous Section.

After an appropriate finite dimensional reduction, in Section 1.6, we findthat stable critical points on S1 × (0,+∞) × R

N of a suitable functional Γcorrespond to points on Z =

eiσzµ,ξ : σ ∈ S1, µ > 0, ξ ∈ R

N

from whichsolutions to (1.3.3) bifurcate for ε 6= 0. The main result of the paper is Theo-rem 1.7.1, stated in Section 1.7, where we prove the existence of a solution uεto (1.3.3) that is close for ε small enough to a solution of (1.3.4). If V changesits sign, we find at least two solutions to (1.3.3). These results are containedin a joint paper with S. Cingolani and S. Secchi [18].Finally, in Sections 1.8 and 1.9, we recall certain regularity properties of solu-tions for a rather general class of equations involving the Schrodinger operatorLA established in [36] which can be apply in our case.

Remark 1.3.1. It is apparent that the compact group S1 acts on the spaceof solutions to (1.3.3). For simplicity, we will talk about solutions, rather thanorbits of solutions.

Notation. The complex conjugate of any number z ∈ C will be denoted byz. The real part of a number z ∈ C will be denoted by Re z. The ordinaryinner product between two vectors a, b ∈ R

N will be denoted by a · b. Weuse the Landau symbols. For example O(ε) is a generic function such that

lim supε→0O(ε)ε < ∞, and o(ε) is a function such that lim

ε→0

o(ε)ε = 0. We

will denote E = D1,2(RN ,C) =u ∈ L2∗(RN ,C) |

∫RN |∇u|2 dx <∞

, with a

similar definition for D1,2(RN ,R).

1.4 The limiting problem

Before proceeding, we recall some known facts about a couple of auxiliaryproblems. Recall that 2∗ = 2N/(N − 2).

(•) The problem −∆u = |u|2

∗−2u in RN

u ∈ D1,2(RN ,R)(1.4.1)

possesses a smooth manifold of least-energy solutions

Z =zµ,ξ = µ−

(N−2)2 z0(

x−ξµ ) : µ > 0, ξ ∈ R

N

(1.4.2)


where

z0(x) = κN1

(1 + |x|2)N−2

2

, κN = (N(N − 2))N−2

4 . (1.4.3)

Explicitly,

zµ,ξ(x) = κNµ− (N−2)

21

(1 +

∣∣∣x−ξµ∣∣∣2)N−2

2

= κNµ(N

2−1)

(µ2 + |x− ξ|2)N−2

2

. (1.4.4)

These solutions are critical points of the Euler functional

f0(u) =1

2

∫

RN

|∇u|2 −1

2∗

∫

RN

u2∗

+ dx, (1.4.5)

defined on D1,2(RN ,R) ⊂ E, and the following nondegeneracy property holds:

ker f ′′0 (zµ,ξ) = Tzµ,ξZ for all µ > 0, ξ ∈ R

N . (1.4.6)

Following the ideas of [70] and [93], we give the following characterization.

Lemma 1.4.1. Any least-energy solution to the problem

−∆u = |u|2

∗−2u in RN

u ∈ D1,2(RN ,C)(1.4.7)

is of the form u = eiσzµ,ξ for some suitable σ ∈ [0, 2π], µ > 0 and ξ ∈ RN .

Proof. It is convenient to divide the proof into two steps.Step 1: Let z0 = U the least energy solution associated to the energy func-tional (1.4.5) on the manifold

M0,r =

v ∈ D1,2(RN ,R) \ 0 :

∫

RN

|∇v|2 dx =

∫

RN

|v|2∗

dx

.

It is well-known that z0 = U is radially symmetric and unique (up to trans-lation and dilation) positive solution to the equation (1.4.1). Let b0,r = br =

f0(U) = f0(z0). In a similar way, we define the class

M0,c =

v ∈ E \ 0 :

∫

RN

|∇v|2 dx =

∫

RN

|v|2∗

dx


and denote by b0,c = bc = f0(v) on M0,c, where

f0(u) =1

2

∫

RN

|∇u|2 −1

2∗

∫

RN

|u|2∗dx,

is the Euler functional associated to (1.4.7). Let σ ∈ R, ξ ∈ RN , µ > 0,

v(x) = zµ,ξ(x) a positive solution to (1.4.1) and U = eiσ v = eiσzµ,ξ (i.e.

zµ,ξ(x) = |U(x)|). It follows that U = eiσzµ,ξ is a non-trivial least energysolution for b0,c = f0(v) with v ∈ M0,c.

Step 2: The following facts hold:

(i) b0,c = b0,r;

(ii) If Uc = U is a least energy solution of problem (1.4.7), then

|∇|Uc|(x)| = |∇Uc(x)| and Re(iUc(x)∇Uc(x)

)= 0 for a.e. x ∈ R

N .

(iii) There exist σ ∈ R and a least-energy solution ur : RN → R of problem

(1.4.1) withUc(x) = eiσur(x) for a.e. x ∈ R

N ,

or, equivalently, the least energy-solution Uc for b0,c is the following

Uc(x) = eiσur(x) = eiσzµ,ξ(x) for a.e. x ∈ RN .

Observe that

b0,r = minv ∈M0,r

f0(v) and b0,c = minv ∈M0,c

f0(v),

where M0,r and M0,c are the real and complex Nehari manifolds for f0 and f0,

M0,r =v ∈ D1,2(RN ,R) \ 0 : f ′0(v)[v] = 0

=

v ∈ D1,2(RN ,R) \ 0 :

∫

RN

|∇v|2 dx =

∫

RN

|v|2∗

dx

and

M0,c =v ∈ E \ 0 | f ′0(v)[v] = 0

=

v ∈ E \ 0 |

∫

RN

|∇v|2 dx =

∫

RN

|v|2∗

dx

.


So (i) is equivalent to

b0,r = minv ∈M0,r

f0(v) = f0(ur),

b0,c = minv ∈M0,c

f0(v) = f0(Uc).

Proof of (i)–(iii). Let u ∈ E be given. For the sake of convenience, weintroduce the functionals

T (u) =

∫

RN

|∇u|2 dx,

P (u) =1

2∗

∫

RN

|u|2∗

dx

(respectively T (u) and P (u) as u ∈ D1,2(RN ,R)) such that f0(u) = 12T (u) −

P (u) as u ∈ E (respectively f0(u) = 12 T (u) − P (u) as u ∈ D1,2(RN ,R)).

Consider the following minimization problems

σr = minT (u) : u ∈ D1,2(RN ,R), P (u) = 1

σc = min T (u) : u ∈ E,P (u) = 1

Note that, obviously, there holds σc ≤ σr. If we denote by u∗ the Schwarzsymmetric rearrangement (see [25]) of the positive real valued function |u| ∈D1,2(RN ,R), then, Cavalieri’s principle yields

∫

RN

|u∗|2∗ dx =

∫

RN

|u|2∗

dx

which entails P (u∗) = P (|u|). Moreover, by the Polya-Szego inequality, wehave

T (u∗) =

∫

RN

|∇u∗|2 dx ≤

∫

RN

|∇|u||2 dx ≤

∫

RN

|∇u|2 dx = T (u),

where the second inequality follows from the following diamagnetic inequality∫

RN

|∇|u||2 dx ≤

∫

RN

|Dε|u||2 dx for all u ∈ HεA,V

with Dε = ∇i − εA and A = 0. Therefore, one can compute σc by minimizing

over the subclass of positive, radially symmetric and radially decreasing func-tions u ∈ D1,2(RN ,R). As a consequence, we have σr ≤ σc. In conclusion,


σr = σc. Observe now that

b0,r = minf0(u) | u ∈ D1,2(RN ,R) \ 0 is a solution to (1.4.1)

,

b0,c = min f0(u) | u ∈ E \ 0 is a solution to (1.4.7) .

The above inequalities hold since any nontrivial real (respectively complex)solution of (1.4.1) (respectively (1.4.7)) belongs to M0,r (respectively M0,c)and, conversely, any solution of b0,r (respectively b0,c) produces a nontrivialsolution of (1.4.1) (respectively (1.4.7)). Moreover, it follows from an easyadaptation of [25, Theorem 3] that b0,r = σr as well as b0,c = σc. In conclusion,there holds

b0,r = σr = b0,c = σc,

which proves (i).

To prove (ii), let Uc : RN → C be a least-energy solution to problem (1.4.7)

and assume by contradiction that

LN( x ∈ R

N : |∇|Uc|| < |∇Uc| )

> 0

where LN is Lebesgue measure in RN . Then, we would get P (|Uc|) = P (Uc)

and

P (|Uc|) =1

2∗

∫

RN

|Uc|2∗ dx =

1

2∗

∫

RN

|Uc|2∗ dx = P (Uc)

and

σr ≤

∫

RN

|∇|Uc||2 dx <

∫

RN

|∇Uc|2 dx = σc

which is a contradiction. The second assertion in (ii) follows by direct com-putations. Indeed, almost everywhere in R

N , we have

|∇|Uc|| = |∇Uc| if and only if ReUc (∇ ImUc) = ImUc∇ (ReUc) .

If this last condition holds, in turn, almost everywhere in RN , we have

Uc∇Uc = ReUc∇ (ReUc) + ImUc∇ (ImUc) ,

which implies the desired assertion.

Finally, the representation formula of (iii) Uc(x) = eiσur(x) is an immedi-ate consequence of (ii), since one obtains Uc = eiσ|Uc| for some σ ∈ R.


By Lemma 1.4.1, it follows that:

(••) f0 ∈ C2(D1,2(RN ,C)) possesses a finite–dimensional manifold Z of least-energy critical points, given by

Z =eiσzµ,ξ : σ ∈ S1, µ > 0, ξ ∈ R

N∼= S1 × (0,+∞) × R

N . (1.4.8)

Remark 1.4.2. For the reader’s convenience, we write here the second deriva-tive of f0 at any z ∈ Z:

⟨f ′′0 (z)v, w

⟩E

= Re

∫

RN

∇v · ∇w dx− Re

∫

RN

|z|2∗−2vw dx

− Re(2∗ − 2)

∫

RN

|z|2∗−4 Re(zv)zw dx. (1.4.9)

In particular, f ′′0 (z) can be identified with a compact perturbation of the iden-tity operator.

We now come to the most delicate requirement of the perturbation method.

Lemma 1.4.3. For each z = eiσzµ,ξ ∈ Z, there holds

TzZ = ker f ′′0 (z) for all z ∈ Z,

where

Teiσzµ,ξZ = spanR

∂eiσzµ,ξ∂ξ1

, . . . ,∂eiσzµ,ξ∂ξN

,∂eiσzµ,ξ∂µ

,∂eiσzµ,ξ∂σ

= ieiσzµ,ξ

.

(1.4.10)

Proof. The inclusion TzZ ⊂ ker f ′′0 (z) is always true, see [6]. Conversely, weprove that for any ϕ ∈ ker f ′′0 (z) there exist numbers a1, . . . , aN , b, d ∈ R suchthat

ϕ =N∑

j=1

aj∂eiσzµ,ξ∂ξj

+ b∂eiσzµ,ξ∂µ

+ dieiσzµ,ξ. (1.4.11)

If we can prove the following representation formulæ, then (1.4.11) will follow.

Re(ϕeiσ) =N∑

j=1

aj∂zµ,ξ∂ξj

+ b∂zµ,ξ∂µ

(1.4.12)

Im(ϕeiσ) = dzµ,ξ. (1.4.13)


We will use a well-known result for the scalar case:

ker f ′′0 (zµ,ξ) ≡ Tzµ,ξZ = spanR

∂zµ,ξ∂ξ1

, . . . ,∂zµ,ξ∂ξN

,∂zµ,ξ∂µ

.

Step 1: Proof of (1.4.12). We wish to prove that Re(ϕeiσ) ∈ ker f ′′0 (zµ,ξ).Recall that ϕ ∈ ker f ′′0 (eiσzµ,ξ), so

〈f ′′0 (eiσzµ,ξ)ϕ,ψ〉 = 0 for all ψ ∈ E. (1.4.14)

Select ψ = eiσv, with v ∈ C∞0 (RN ,R).

0 = 〈f ′′0 (eiσzµ,ξ)ϕ, veiσ〉 = Re

∫∇(ϕe−iσ)∇v

− (2∗ − 2)

∫

RN

|zµ,ξ|2∗−2 Re(eiσϕ)v −

∫

RN

|zµ,ξ|2∗−2 Re(eiσϕ)v

=

∫

RN

∇(Re(ϕeiσ)∇v − (2∗ − 1)

∫

RN

|zµ,ξ|2∗−2 Re(eiσϕ)v

= 〈f ′′0 (zµ,ξ)Re(ϕeiσ), v〉.

This implies that

Re(eiσϕ) ∈ ker f ′′0 (zµ,ξ) ≡ Tzµ,ξZ

from which it follows

Re(ϕeiσ) =N∑

j=1

aj∂zµ,ξ∂ξj

+ b∂zµ,ξ∂µ

for some real constants a1, . . . , aN and b.Step 2: Proof of (1.4.13). Test (1.4.14) on ψ = ieiσw ∈ E with w :

RN → R. We get

0 = 〈f ′′0 (eiσzµ,ξ)ϕ, ieiσw〉 = Re

∫

RN

∇(−iϕe−iσ) · ∇w

− Re

∫

RN

|zµ,ξ|2∗−2(−iϕe−iσ)w

[since Re(−iϕe−iσ) = Im(ϕe−iσ)]

=

∫

RN

∇(Im(ϕe−iσ)) · ∇w −

∫

RN

|zµ,ξ|2∗−2 Im(ϕe−iσ)w

=

∫

RN

∇(Im(ϕeiσ)) · ∇w −

∫

RN

|zµ,ξ|2∗−2

[Im(ϕeiσ)

]+w.

(1.4.15)


We can take µ = 1 and ξ = 0, otherwise we perform the change of variablex 7→ µx+ ξ.

From (1.4.15) we get that u := Im(ϕeiσ) satisfies the equation

−∆u =N(N − 2)

(1 + |x|2)2u in D−1,2(RN ,R). (1.4.16)

We will study this linear equation by an inverse stereographic projections ontothe sphere SN . Precisely, for each point ξ ∈ SN , denote by x its correspondingpoint under the stereographic projection π from SN to R

N , sending the northpole on SN to ∞. That is, suppose ξ = (ξ1, ξ2, . . . , ξN+1) is a point in SN ,

x = (x1, . . . , xN ), then ξi = 2xi

1+|x|2 for 1 ≤ i ≤ N ; ξN+1 = |x|2−1|x|2+1

.

Recall that, on a Riemannian manifold (M, g), the conformal Laplacian isdefined by

Lg = −∆g +N − 2

4(N − 1)Sg,

where −∆g is the Laplace–Beltrami operator on M and Sg is the scalar cur-vature of (M, g). It is known that

Lg(Φ(u)) = ϕ−N+2N−2Lδ(u),

where δ is the Euclidean metric of RN , ϕ(x) =

(2

1+|x|2

)(N−2)/2and

Φ: D1,2(RN ) → H1(SN ), Φ(u)(x) =u(π(x))

ϕ(π(x))

is an isomorphism between H1(SN ) and E := D1,2(RN ). Therefore, if U =Φ(u), then (1.4.16) changes into the equation

−∆g0U +N − 2

4(N − 1)Sg0U =

N(N − 2)

4U, (1.4.17)

where g0 is the standard Riemannian metric on SN , and Sg0 = N(N − 1) isthe constant scalar curvature of (SN , g0). As a consequence, (1.4.17) impliesthat

−∆g0U = 0,

i.e. U is an eigenfunction of −∆g0 corresponding to the eigenvalue λ = 0. Butthe point spectrum of −∆g0 is completely known (see [26, 27]), consisting ofthe numbers

λk = k(k +N − 1), k = 0, 1, 2, . . .

1.5. The variational framework 25

with associated eigenspaces of dimension

(N + k − 2)! (N + 2k − 1)

k! (N − 1)!.

Hence we deduce that k = 0, and U belongs to an eigenspace of dimension 1.Since zµ,ξ is a solution to (1.4.16), we conclude that there exists d ∈ R suchthat

Im(ϕeiσ) = dzµ,ξ.

This completes the proof.

1.5 The variational framework

In the variational framework of the problem, solutions to (1.3.3) can be foundas critical points of the energy functional fε : E → R defined by

fε(u) =1

2

∫

RN

∣∣∣∣(∇

i− εA(x)

)u

∣∣∣∣2

dx+εα

2

∫

RN

V (x)|u|2dx−1

2∗

∫

RN

|u|2∗dx,

(1.5.1)on the real Hilbert space

E = D1,2(RN ,C) =

v ∈ L2∗(RN ,C) :

∫

RN

|∇v|2dx <∞

(1.5.2)

endowed with the inner product

〈u, v〉E = Re

∫

RN

∇u · ∇v dx. (1.5.3)

We shall assume throughout the paper that

(N) N > 4,

(A1) A ∈ C1(RN ,RN ) ∩ L∞(RN ,RN ) ∩ Lr(RN ,RN ) with 1 < r < N ,

(A2) divA ∈ LN/2(RN ,R),

(V) V ∈ C(RN ,R) ∩ L∞(RN ,R) ∩ Ls(RN ,R) with 1 < s < N/2.


The functional fε is well defined on E. Indeed,

∫

RN

∣∣∣∣(∇

i− εA(x)

)u

∣∣∣∣2

=

∫

RN

|∇u|2 + ε2∫

RN

|A|2|u|2 − Re

∫

RN

∇u

i· εAu,

and all the integrals are finite by virtue of (A1). Moreover, fε ∈ C2(E,R).

In this Section, we apply Lemmma 1.1.4 and Lemma 1.2.2 to performa finite–dimensional reduction on fε according to the methods of [6, 8, 10]presented in Sections 1.1 and 1.2. Roughly speaking, since the unperturbedproblem (i.e. (1.3.3) with ε = 0) has a whole C2 manifold of critical points,we can deform this manifold is a suitable manner and get a finite–dimensionalnatural constraint for the Euler–Lagrange functional associated to (1.3.3). Asa consequence, we can find solutions to (1.3.3) in correspondence to (stable)critical points of an auxiliary map — called the Melnikov function — in finitedimension.

Now we focus on the case α = 2, as in the other cases α ∈ [1, 2) themagnetic potential A no longer affects the finite-dimensional reduction (seeRemark (1.7.2)).

Firstly, we can write the functional fε as

fε(u) = f0(u) + εG1(u) + ε2G2(u), (1.5.4)

where

f0(u) =1

2

∫

RN

|∇u|2 −1

2∗

∫

RN

|u|2∗, (1.5.5)

G1(u) = −Re1

i

∫

RN

∇u ·Au, G2(u) =1

2

∫

RN

|A|2|u|2 +1

2

∫

RN

V (x)|u|2.

(1.5.6)So that fε has the form (1.2.2) given in Section 1.2 with f0, G1, G2 ∈ C2(E,R).By (••) in Section 1.4, it follows that hypothesis (CM) is satisfied, i.e. Z =eiσzµ,ξ : σ ∈ S1, µ > 0, ξ ∈ R

N

defined in (1.4.8) is a finite dimensional man-ifold and every z ∈ Z is a critical point of f0. Assumption (Fr) holds too, sincef ′′0 (z) = I − C, C compact for every z ∈ Z, and (ND) follows from Lemma1.4.3. Then, we can apply Lemma 1.1.4 to build a natural constraint Zε forthe functional fε. Furthermore, one has

Lemma 1.5.1. G1(z) = 0 for all z ∈ Z.


Proof.

G1(z) = −Re

∫

RN

∇z

i·A(x) z dx =

[z = eiσzµ,ξ

]

= −Re

∫

RN

eiσ∇zµ,ξi

·A(x) e−iσzµ,ξ dx =

= −Re

∫

RN

∇zµ,ξi

·A(x) zµ,ξ dx = 0.

Using Lemma 1.2.2, we can find the Melnikov function defined in (1.2.4).

Remark 1.5.2. We notice that Γ = G2(z) + 12 (G′

1(z), φ), where z stands foreiσzµ,ξ and φ = limε→0

wε .

Remark 1.5.3. By the definition of z ∈ Z, it follows that: Γ(z) = Γ(eiσzµ,ξ) =Γ(σ, µ, ξ). In the sequel, we will write freely Γ(σ, µ, ξ) ≡ Γ(µ, ξ) since Γ is σ-invariant. Indeed, it is easy to check that G2 is σ-invariant. In fact, by thedefinition of G2(z) and z = eiσzµ,ξ, it follows that:

G2(σ, µ, ξ) = G2(eiσzµ,ξ) =

1

2

∫

RN

|A(x)|2|zµ,ξ|2 dx+

1

2

∫

RN

V (x)|zµ,ξ|2 dx

≡ G2(µ, ξ).

It remains to prove that 〈G′1(z), φ〉 is σ-invariant. We will show that φ =

eiσψ(µ, ξ) with ψ(µ, ξ) ∈ C independent of σ which immediately gives

⟨G′

1(eiσzµ,ξ), φ

⟩= −Re

∫

RN

1

ieiσ∇zµ,ξ ·A(x)e−iσψ(µ, ξ) dx

−Re

∫

RN

1

i∇ψµ,ξ ·A(x)zµ,ξ dx =

⟨G′

1(zµ,ξ), ψ(µ, ξ)⟩.

We begin to recall that φ = limε→0+w(ε,z)ε , where w(ε, z) is such that

f ′ε(eiσzµ,ξ + w(σ, µ, ξ)) ∈ Teiσzµ,ξ

Z.

By (1.4.10), this condition means that

f ′ε(eiσzµ,ξ + w(σ, µ, ξ)) =

N∑

i=1

aieiσ ∂zµ,ξ∂ξi

+ beiσ∂zµ,ξ∂µ

+ deiσizµ,ξ, (1.5.7)


with a1, . . . , aN , b, d, ∈ R.

Let w(σ, µ, ξ) = eiσw with w ∈ D1,2(RN ,C). Testing (1.5.7) by eiσv(x)with v ∈ D1,2(RN ,C), we derive that zµ,ξ + w is a solution of an equationindependently of σ. Thus, also w is independent of σ and it can be denotedas w(µ, ξ). Set ψ(µ, ξ) = limε→0+

w(µ,ξ)ε , we deduce that φ = eiσψ(µ, ξ).

1.6 Asymptotic study of Γ

In order to find critical points of Γ it is convenient to study the behavior of Γas µ→ 0 and as µ+ |ξ| → ∞. Our goal is to show:

Proposition 1.6.1.Γ can be extended smoothly to the hyperplane

(0, ξ) ∈ R × R

N

by setting

Γ(0, ξ) = 0. (1.6.1)

Moreover, it follows that

Γ(µ, ξ) → 0, as µ+ |ξ| → +∞. (1.6.2)

The proof of this Proposition is rather technical, so we split it into severallemmas in which we will use the formulation of Γ = G2(z)+

12 (G′

1(z), φ), whereφ = limε→0

wε .

Lemma 1.6.2. Under assumption (A1) there holds

limµ→0+

1

2

∫

RN

|A(x)|2|zµ,ξ|2dx = 0. (1.6.3)

Proof. Let z = eiσzµ,ξ ∈ Z. Then

H2(z) =1

2

∫

RN

|A(x)|2|zµ,ξ|2dx

=1

2

∫

RN

|A(x)|2

κNµ−

(N−2)2

(1 +

∣∣∣∣x− ξ

µ

∣∣∣∣2) 2−N

2

2

dx

=κ2N

2µ(N−2)

∫

RN

|A(x)|2(1 +

∣∣x−ξµ

∣∣2)N−2dx. (1.6.4)

1.6. Asymptotic study of Γ 29

Using the change of variable y = x−ξµ , or x = µy + ξ, we can write

H2(z) =κ2N

2µ(N−2)

∫

RN

|A(µy + ξ)|21

(1 + |y|2)N−2µNdy

=κ2N

2µ2

∫

RN

|A(µy + ξ)|2

(1 + |y|2)N−2dy

and using the hypothesis (A1)

H2(z) ≤ µ2CN‖A‖2∞

∫

RN

1

(1 + |y|2)N−2dy, (1.6.5)

the claim follows.

The proof of the following Lemma is similar and thus omitted.

Lemma 1.6.3. Under assumption (V) there holds

limµ→0+

1

2

∫

RN

V (x)|zµ,ξ|2dx = 0. (1.6.6)

Lemma 1.6.4. There holds

limµ→0+

〈G′1(z), φ〉 = 0. (1.6.7)

Proof. We write 〈G′1(z), φ〉E = α1 + α2, where

α1 = −Re

∫

RN

∇z

i·A(x)φdx (1.6.8)

α2 = −Re

∫

RN

∇φ

i·A(x)z dx. (1.6.9)

It is convenient to introduce φ∗(y) by setting

φ∗(y) = φ∗µ,ξ(y) = µN2−1φ(µy + ξ).

Using the expression of z = eiσµ−(N−2)

2 z0(x−ξµ ) and the change of variable

x = µy + ξ we can write:

α1 = −Re

∫

RN

1

i∇xe

iσµ−(N−2)

2 z0

(x− ξ

µ

)·A(x)φ(x) dx

= −Re

∫

RN

1

ieiσ∇yz0(y)µ

N2 ·A(µy + ξ)φ(µy + ξ) dy

= −µRe

∫

RN

1

ieiσ∇yz0(y) ·A(µy + ξ)φ∗(y) dy


and

α2 = −Re

∫

RN

1

i∇xφ(x) ·A(x)e−iσµ−

(N−2)2 z0

(x− ξ

µ

)dx

= −Re

∫

RN

1

i∇yφ(µy + ξ)µ−1 ·A(µy + ξ)e−iσµ−

(N−2)2 µNz0(y) dy

= −Re

∫

RN

1

i∇φ(µy + ξ)µ−1 ·A(µy + ξ)e−iσµ

N2

+1z0(y) dy

= −µRe

∫

RN

1

i∇φ∗(y) ·A(µy + ξ)e−iσz0(y) dy.

Now the conclusion follows easily from the next Lemma.

Lemma 1.6.5. As µ→ 0+,

φ∗µ,ξ → 0 strongly in E. (1.6.10)

Proof. For all v ∈ E, due to the divergence theorem, we have

⟨G′

1(z), v⟩E

= −Re

∫

RN

∇z

i·A(x)v dx− Re

∫

RN

∇v

i·A(x)z dx

= −Re

∫

RN

∇z

i·A(x)v dx− Re

∫

RN

1

i

N∑

j=1

∂v

∂xjAj(x)z dx

= −Re

∫

RN

∇z

i·A(x)v dx+ Re

∫

RN

1

i

N∑

j=1

v∂

∂xj(Ajz) dx

= −Re

∫

RN

∇z

i·A(x)v dx+ Re

∫

RN

1

iv divAz dx+ Re

∫

RN

1

ivA · ∇z dx

= −2 Re

∫

RN

∇z

i·A(x)v dx− Re

∫

RN

1

idivAzv dx,

where the last integral is finite by assumption (A2) and

(f ′′0 (z)wµ,ξ, v) = Re

∫

RN

∇wµ,ξ · ∇v dx− Re

∫

RN

|z|2∗−2wµ,ξv dx

− Re

∫

RN

(2∗ − 2)|z|2∗−4 Re(zwµ,ξ)zv dx. (1.6.11)

We know that wµ,ξ = −εLeiσzµ,ξG′

1(eiσzµ,ξ) + o(ε), and hence

〈f ′′0 (z)φµ,ξ, v〉E = −〈G′1(z), v〉E , ∀v ∈ E, (1.6.12)


where φµ,ξ = limǫ→0wµ,ξ

ǫ . This implies that φµ,ξ solves

Re

∫

RN

∇φµ,ξ · ∇vdx− Re

∫

RN

|z|2∗−2φµ,ξv dx

− Re

∫

RN

(2∗ − 2)|z|2∗−4 Re(zφµ,ξ)zv dx

= 2 Re

∫

RN

1

i∇z ·A(x)vdx+ Re

∫

RN

1

idivAzv dx.

Multiplying by µN2−1 and using the expression of z = eiσµ−

(N−2)2 z0(

x−ξµ ), we

get

Re

∫

RN

µN2−1∇xφµ,ξ(x)∇vdx

− Re

∫

RN

µ−2

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗−2

µN2−1φµ,ξ(x)vdx

− Re

∫

RN

(2∗ − 2)µN−4

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗−4

× Re

(eiσµ−

N2

+1z0

(x− ξ

µ

)µ

N2−1φµ,ξ(x)

)eiσµ−

N2

+1z0

(x− ξ

µ

)vdx

= 2 Re

∫

RN

1

ieiσ∇xz0

(x− ξ

µ

)·A(x)vdx

+ Re

∫

RN

1

idivAeiσz0

(x− ξ

µ

)vdx.

Using the expression of φ∗(x−ξµ ) = µN2−1φµ,ξ(x), we have

Re

∫

RN

∇xφ∗

(x− ξ

µ

)∇vdx−Re

∫

RN

µ−2

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗−2

φ∗(x− ξ

µ

)vdx

− Re

∫

RN

(2∗ − 2)µN−4

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗−4

Re

(eiσµ−

N2

+1z0

(x− ξ

µ

)φ∗(x− ξ

µ

))× eiσµ−

N2

+1z0

(x− ξ

µ

)vdx

= 2 Re

∫

RN

1

ieiσ∇xz0

(x− ξ

µ

)·A(x)vdx

+ Re

∫

RN

1

idivA(x) e−iσz0

(x− ξ

µ

)vdx;


then, the change of variable x = µy + ξ yields

Re

∫

RN

µ−2∇yφ∗(y)∇yv(µy + ξ)µNdy

−Re

∫

RN

µN−2 | z0(y) |2∗−2 φ∗(y)v(µy + ξ)dy−Re

∫

RN

(2∗−2)µN−4 | z0(y)|2∗−4

Re(eiσµ2(−N

2+1)z0(y)φ∗(y)

)eiσz0(y)v(µy + ξ)µNdy

= 2 Re

∫

RN

1

ieiσ∇yz0(y) ·A(µy + ξ)v(µy + ξ)µN−1dy

+ Re

∫

RN

1

idivA(µy + ξ) e−iσz0(y)v(µy + ξ)µNdy.

Replacing x = y and dividing by µN−2, it follows that

Re

∫

RN

∇xφ∗(x)∇xv(µx+ ξ) dx− Re

∫

RN

| z0(x) |2∗−2 φ∗(x)v(µx+ ξ) dx

− Re

∫

RN

(2∗ − 2) | z0(x)|2∗−4 Re

(eiσz0(x)φ

∗(x))eiσz0(x)v(µx+ ξ) dx

= 2µRe

∫

RN

1

ieiσ∇xz0(x) ·A(µx+ ξ))v(µx+ ξ) dx

+ µ2 Re

∫

RN

1

idivy A(µx+ ξ) eiσz0(x)v(µx+ ξ) dx.

This means that, if we write τµ,ξ(x) = µx+ ξ,

⟨f ′′0 (eiσz0)φ

∗, v τµ,ξ⟩

=

∫

RN

kµ,ξv τµ,ξ

for all test function v; in particular,

f ′′0 (eiσz0)φ∗ = kµ,ξ

where

kµ,ξ(x) =2

iµeiσ∇xz0(x) ·A(µx+ ξ) +

1

iµ2eiσ divy A(µx+ ξ) z0(x).

We conclude that φ∗ is a solution of

φ∗(x) = Leiσz0kµ,ξ(x). (1.6.13)


Our assumptions on A (i.e. (A1) and (A2)) imply immediately that

kµ,ξ → 0 in E as µ→ 0. (1.6.14)

From the continuity of Leiσz0 we deduce that

limµ→0+

φ∗ = limµ→0+

Leiσz0kµ,ξ = 0. (1.6.15)

This completes the proof of the Lemma.

Lemma 1.6.6. Under assumption (A1), there holds

limµ+|ξ|→+∞

H2(µ, ξ) = 0,

where H2 is defined in (1.6.4).

Proof. Firstly, assume that µ → µ ∈ (0,+∞) and µ + |ξ| → +∞. We noticethat

H2(µ, ξ) =µ−(N−2)

2

∫

RN

|A(x)|2z20

(x− ξ

µ

)dx

=µ−(N−2)

2

∫

|x|≤ |ξ|2

|A(x)|2z20

(x− ξ

µ

)dx

+µ−(N−2)

2

∫

|x|> |ξ|2

|A(x)|2z20

(x− ξ

µ

)dx.

Moreover,

µ−(N−2)

2

∫

|x|≤ |ξ|2

|A(x)|2z20

(x− ξ

µ

)dx

≤µ−(N−2)

2||A||2∞ωN

|ξ|N

2Nsup

|x|≤ |ξ|2

z20

(x− ξ

µ

)

=µ−(N−2)

2||A||2∞ωN

|ξ|N

2Nsup

|x|≤ |ξ|2

k2Nµ

2(N−2)

[µ2 + |x− ξ|2]N−2

≤µ−(N−2)

2||A||2∞ωN

|ξ|N

2Nsup

|x|≤ |ξ|2

k2N[

µ2 + | |x| − |ξ| |2]N−2

≤µ−(N−2)

2||A||2∞ωN

|ξ|N

2Nk2N[

µ2 + |ξ|2

4

]N−2,


where ωN is the measure of SN−1 =x ∈ R

N : |x| = 1. Since N > 4, we

inferk2N |ξ|

N

[µ2 + |ξ|2

4

]N−2→ 0 as |ξ| → +∞.

Finally, we deduce

µ−(N−2)

2

∫

|x|≤ |ξ|2

|A(x)|2z20

(x− ξ

µ

)dx→ 0

as µ→ µ and |ξ| → +∞. On the other hand, we have

µ−(N−2)

2

∫

|x|> |ξ|2

|A(x)|2z20

(x− ξ

µ

)dx

≤µ−(N−2)

2||A||2∞

∫

|x|> |ξ|2

z20

(x− ξ

µ

)dx

=µN−(N−2)

2||A||2∞

∫

|µx+ξ|> |ξ|2

z20(x)dx.

Since z20 ∈ L1(RN ), we deduce that

µ2

2||A||2∞

∫

|µx+ξ|> |ξ|2

z20(x)dx→ 0

and thus

µ−(N−2)

2

∫

|x|> |ξ|2

|A(x)|2z20

(x− ξ

µ

)dx→ 0

as µ→ µ and |ξ| → +∞.Finally, we can conclude that H2(µ, ξ) → 0 as µ→ µ and |ξ| → +∞.Conversely, assume that µ → +∞. After a suitable change of variable, itresults

H2(µ, ξ) =µ2

2

∫

RN

|A(µy + ξ)|2|z0(y)|2dy.

By assumption (A1), we can fix 1 < r < N2 such that A2 ∈ Lr(RN ). Moreover,

let be s = rr−1 . It is immediate to check that 2s > 2∗ and then z2s

0 ∈ L1(RN ).


By (A1) and Holder inequality, we deduce that

∫

RN

|A(µy + ξ)|2|z0(y)|2dy

≤

(∫

RN

|A(µy + ξ)|2rdy

) 1r(∫

RN

|z0(y)|2sdy

) 1s

≤ µ−Nr

(∫

RN

|A(y)|2rdy

) 1r(∫

RN

|z0(y)|2sdy

) 1s

.

As a consequence, by the above inequality, we infer for µ small

G2(µ, ξ) =µ2

2

∫

RN

|A(µy + ξ)|2|z0(y)|2dy

≤ µ2−Nr

(∫

RN

|A(y)|2rdy

) 1r(∫

RN

|z0(y)|2sdy

) 1s

.

Now, we notice that r < N2 implies 2− N

r < 0 and thus by the above inequalitywe can conclude that G2(µ, ξ) tends to 0 as µ→ +∞.

Arguing as before we can deduce the following result.

Lemma 1.6.7. Under assumption (V), there holds

limµ+|ξ|→+∞

∫

RN

V (x)|zµ,ξ(x)|2 dx = 0.

In order to describe the behavior of the term 〈G′1(z), φ〉E as µ+ |ξ| → +∞,

we need the following Lemma.

Lemma 1.6.8. There is a constant CN > 0 such that

‖φ ‖E ≤ CN for all µ > 0 and for all ξ ∈ RN . (1.6.16)

Proof. We know that for all ε > 0 and all z ∈ Z

w(ε, z) = −εLzG′1(z) + o(ε)

so that

φ = limε→0

w(ε, z)

ε= −LzG

′1(z)


and

‖φ‖E ≤ ‖Lz‖∥∥G′

1(z)∥∥ .

We claim that ‖Lz‖ is bounded above by a constant independent of µ and ξ.Indeed:

‖Lz‖ = sup‖ϕ‖=1

‖Lzϕ‖ = sup‖ϕ‖=1‖ψ‖=1

|〈Lzϕ,ψ〉|

= sup‖ϕ‖=1‖ψ‖=1

∣∣∣∣∫

RN

∇ϕ · ∇ψ − Re

∫

RN

|zµ,ξ|2∗−2 ϕψ

− (2∗ − 2)

∫

RN

|zµ,ξ|2∗−4 Re(ϕzµ,ξ)Re(ψzµ,ξ)

∣∣∣∣

≤ sup‖ϕ‖=1‖ψ‖=1

(∫

RN

|∇ϕ|∣∣∇ψ

∣∣+ Re

∫

RN

|zµ,ξ|2∗−2 |ϕ|

∣∣ψ∣∣

+ (2∗ − 2)

∫

RN

|zµ,ξ|2∗−2 |ϕ|

∣∣ψ∣∣)

≤ sup‖ϕ‖=1‖ψ‖=1

(∫

RN

|∇ϕ|∣∣∇ψ

∣∣+ (2∗ − 1)

∫

RN

|zµ,ξ|2∗−2 |ϕ|

∣∣ψ∣∣)

≤ sup‖ϕ‖=1‖ψ‖=1

(∫

RN

|∇ϕ|2)1/2(∫

RN

|∇ψ|2)1/2

+ (2∗ − 1)

(∫

RN

|zµ,ξ|2∗)(2∗−2)/2∗

×

(∫

RN

|ϕ|2∗

)1/2∗ (∫

RN

|ψ|2∗

)1/2∗

.

We observe that

(∫

RN

|zµ,ξ|2∗)1/2∗

= µ−(N−2)

2

(∫

RN

∣∣∣∣z0(x− ξ

µ

)∣∣∣∣2∗)1/2∗

= µ−(N−2)

2

(∫

RN

|z0(y)|2∗ µN

)1/2∗

= ‖z0‖L2∗ .


Hence

‖Lz‖ ≤ sup‖ϕ‖=1‖ψ‖=1

(1 + (2∗ − 1)‖ z0‖

(2∗−2)

L2∗ ‖ϕ‖L2∗‖ψ‖L2∗

)

≤ sup‖ϕ‖=1‖ψ‖=1

(1 + (2∗ − 1)C ′

N‖z0‖(2∗−2)E ‖ϕ‖E‖ψ‖E

)

≤ 1 + (2∗ − 1)C ′N‖z0‖

(2∗−2)E ≡ C1

N ,

where C1N is a constant independent of µ and ξ. At this point it follows that:

‖φ‖ ≤ C1N

∥∥G′1(z)

∥∥

and we have to evaluate ‖G′1(z)‖ :

∥∥G′1(z)

∥∥ = sup‖ϕ‖=1

∣∣⟨G′1(z), ϕ

⟩∣∣

= sup‖ϕ‖=1

∣∣∣∣(−Re

∫

RN

∇z

i·A(x)ϕdx− Re

∫

RN

∇φ

i·A(x)z dx

)∣∣∣∣

≤ sup‖ϕ‖=1

(∫

RN

|∇zµ,ξ| |A(x)| |ϕ| dx+

∫

RN

|∇ϕ| |A(x)| |zµ,ξ| dx

)

≤ ‖A‖LN sup‖ϕ‖=1

(‖z0‖E ‖ϕ‖EC

′′N

)

≤ ‖A‖LN‖z0‖EC′′N ≡ C2

N

with C2N independent of µ and ξ. Finally,

‖φ ‖ ≤ C1NC

2N ≡ CN

with CN independent of µ and ξ and the lemma is proved.

Remark 1.6.9. It is easy to check that ‖φ∗‖ = ‖φ‖.

Lemma 1.6.10. There holds

limµ+|ξ|→+∞

⟨G′

1(z), φ⟩E

= 0.

Proof. Firstly, assume that µ → µ ∈ (0,+∞) and µ + |ξ| → +∞. We canwrite 〈G′

1(z), φ〉E = α1 + α2 where

α1 = −Re

∫

RN

∇z

i·A(x)φdx (1.6.17)

α2 = −Re

∫

RN

∇φ

i·A(x)z dx. (1.6.18)


Using the expression of z = eiσµ−(N−2)

2 z0(x−ξµ ) and by assumption (A1) and

the Holder inequality we have:

α1 = −Re

∫

RN

1

i∇xe

iσµ−(N−2)

2 z0

(x− ξ

µ

)·A(x)φ(x) dx

≤ µ−(N−2)

2

(∫

RN

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

dx

)1/2(∫

RN

(|A(x)| |φ|

)2dx

)1/2

≤ µ−(N−2)

2 ‖A‖LN (RN ) ‖φ‖L2∗ (RN )

(∫

RN

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

dx

)1/2

.

We notice that∫

RN

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

dx =

∫

|x| ≤ |ξ|/2

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

dx

+

∫

|x|> |ξ|/2

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

dx

and ∣∣∣∇xz0

(x−ξµ

)∣∣∣2

= µ2(2−N)(2 −N)2κ2N

|x− ξ|2

(µ2 + |x− ξ|2)N.

Moreover, setting C2N := (2 −N)2κ2

N ,

∫

|x| ≤ |ξ|/2

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

dx ≤ ωN|ξ|N

2Nsup

|x| ≤ |ξ|/2

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

= ωN|ξ|N

2Nsup

|x| ≤ |ξ|/2µ2(2−N)(2 −N)2κ2

N

|x− ξ|2

(µ2 + |x− ξ|2)N

= µ2(2−N)ωN|ξ|N

2Nsup

|x| ≤ |ξ|/2

C2N |x− ξ|2

(µ2 + |x− ξ|2)N

≤ µ2(2−N)ωN|ξ|N

2Nsup

|x| ≤ |ξ|/2

C2N ( |x| + |ξ| )2

(µ2 + | |x| − |ξ| |2

)N

≤9

4ωN

|ξ|N

2NC2N |ξ|2

(µ2 + |ξ|2/4)N.

From N > 4, we infer

C2N |ξ|N+2

(µ2 + |ξ|2/4)N→ 0 as |ξ| → +∞.


Finally, we deduce

∫

|x| ≤ |ξ|/2

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

dx → 0 as µ→ µ ∈ (0,+∞), |ξ| → +∞.

On the other hand, we have

∫

|x|> |ξ|/2

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

dx ≤ µN−2

∫

|µx+ξ|> |ξ|/2|∇xz0(x)|

2 dx.

Since |∇xz0|2 ∈ L1(RN ), we deduce that

µN−2

∫

|µx+ξ|> |ξ|/2|∇xz0(x)|

2 dx → 0

as µ→ µ ∈ (0,+∞) and |ξ| → +∞ and thus

∫

|x|> |ξ|/2

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

dx → 0

and

α1 ≤ µ−(N−2)

2 ‖A‖LN (RN ) ‖φ‖L2∗ (RN )

(∫

RN

∣∣∣∣∇xz0

(x− ξ

µ

)∣∣∣∣2

dx

)1/2

→ 0

as µ→ µ ∈ (0,+∞) and |ξ| → +∞.As regards α2 we know that

α2 = −Re

∫

RN

1

i∇xφ(x) ·A(x)e−iσµ−

(N−2)2 z0

(x− ξ

µ

)dx

≤ µ−(N−2)

2

(∫

RN

| ∇xφ(x) ·A(x)|β dx

)1/β(∫

RN

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗

dx

)1/2∗

≤ µ−(N−2)

2 ‖φ ‖E ‖A ‖LN

(∫

RN

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗

dx

)1/2∗

with β = 2N/(N + 2). We notice that

∫

RN

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗

dx =

∫

|x| ≤ |ξ|/2

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗

dx

+

∫

|x|>|ξ|/2

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗

dx.


Moreover,

∫

|x| ≤ |ξ|/2

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗

dx ≤ ωN|ξ|N

2Nsup

|x| ≤ |ξ|/2

∣∣∣∣z0(x− ξ

µ

)∣∣∣∣2∗

= ωN|ξ|N

2Nsup

|x| ≤ |ξ|/2µ2Nκ2∗

N

µ2Nκ2∗

N

(µ2 + |x− ξ|2)N

≤ µ2NωN|ξ|N

2Nsup

|x| ≤ |ξ|/2

κ2∗

N(µ2 + | |x| − |ξ| |2

)N

≤ µ2NωN|ξ|N

2Nκ2∗

N

(µ2 + |ξ|2/4)N.

From N > 4, we infer

κ2∗

N |ξ|N

(µ2 + |ξ|2/4)N→ 0 as |ξ| → +∞.

Finally, we deduce

∫

|x| ≤ |ξ|/2

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗

dx → 0

as µ→ µ ∈ (0,+∞) and |ξ| → +∞.On the other hand, we have

∫

|x|> |ξ|/2

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗

dx ≤ µN∫

|µx+ξ|> |ξ|/2| z0(x) |

2∗ dx.

Since | z0|2∗ ∈ L1(RN ), we deduce that

µN∫

|µx+ξ|> |ξ|/2| z0(x) |

2∗ dx → 0

as µ→ µ ∈ (0,+∞) and |ξ| → +∞ and thus

∫

|x|> |ξ|/2

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗

dx → 0

and

α2 ≤ µ−(N−2)

2 ‖A‖LN (RN ) ‖φ‖E

(∫

RN

∣∣∣∣ z0(x− ξ

µ

) ∣∣∣∣2∗

dx

)1/2∗

→ 0


as µ→ µ ∈ (0,+∞) and |ξ| → +∞.Conversely, assume that µ → +∞. Now it is convenient to write

⟨G′

1(z), φ⟩E

= α1 + α2,

where

α1 = −µRe

∫

RN

eiσ

i∇yz0(y) ·A(µy + ξ)φ∗(y) dy

and

α2 = −µRe

∫

RN

1

i∇yφ

∗(y) ·A(µy + ξ)e−iσz0(y) dy.

The Holder inequality implies that

α1 ≤ µ‖φ∗ ‖L2∗

(∫

RN

(∇yz0(y) ·A(µy + ξ))β dy

)1/β

,

where 1/2∗ + 1/β = 1 so β = 2N/(N + 2). By assumptions (A1), we can fixr ∈ (1, (N + 2)/2) such that Aβ ∈ Lr(RN ). Moreover, let s = r/(r − 1). It isimmediate to check that βs > 2 and then |∇yz0|

βs ∈ L1(RN ). By (A1) andthe Holder inequality, we deduce that:

(∫

RN

(∇yz0(y) ·A(µy + ξ))β dy

)1/β

≤

(∫

RN

(∇yz0(y))βs dy

)1/βs(∫

RN

(A(µy + ξ))βr dy

)1/βr

≤ µ− N

βr ‖∇yz0(y)‖Lβs

(∫

RN

(A(µy + ξ))βr dy

)1/βr

.

As a consequence, by the above inequality, we infer for µ small:

α1 ≤ µ1− N

βr ‖∇yz0(y)‖Lβs

(∫

RN

(A(µy + ξ))βr dy

)1/βr

‖φ∗ ‖L2∗

≤ µ1− N

βrC ′N‖z0‖E‖A‖Lβr‖φ∗‖E .

Analogously,

α2 ≤ µ

(∫

RN

( |∇yφ∗(y)| |A(µy + ξ)| dy )β

)1/β(∫

RN

|z0(y)|2∗ dy

)1/2∗

≤ µ1− N

βr ‖ z0 ‖L2∗

(∫

RN

(∇yφ∗(y))βs dy

)1/βs(∫

RN

(A(y))βr dy

)1/βr

≤ µ1− NβrC ′′

N (‖ z0 ‖E ‖A ‖Lβr ‖φ∗ ‖E) .


Since β = 2N/(N + 2), we deduce 1 − Nβr < 0. The conclusion follows imme-

diately from Lemma 1.6.8.

Proposition 1.6.11. Assume that there exists ξ ∈ RN with V (ξ) 6= 0. Then

limµ→ 0+

Γ(µ, ξ)

µ2=

1

2V (ξ)

∫

RN

|z0|2. (1.6.19)

In particular, Γ is a non-constant map.

Proof. If V (ξ) 6= 0 for some ξ ∈ RN , we can immediately check that Γ(µ, ξ)

is not identically zero. More precisely, we prove that for every ξ ∈ RN there

holds

limµ→ 0+

Γ(µ, ξ)

µ2=

1

2V (ξ)

∫

RN

|z0|2. (1.6.20)

Indeed, after a suitable change of variable,

limµ→ 0+

G2(zµ,ξ)

µ2=

limµ→ 0+

1

2

∫

RN

|A(µy + ξ)|2|z0(y)|2 +

1

2

∫

RN

V (µy + ξ)|z0(y)|2 dy

=1

2|A(ξ)|2

∫

RN

|z0(y)|2 dy +

1

2V (ξ)

∫

RN

|z0(y)|2 dy. (1.6.21)

To complete the proof of (1.6.20), we need to study

limµ→ 0+

1

2µ2

⟨G′

1(zµ,ξ), φµ,ξ⟩.

In Lemma 1.6.5, we have showed that

⟨G′

1(eiσzµ,ξ), φµ,ξ

⟩= −〈f ′′0 (zµ,ξe

iσ)φµ,ξ, φµ,ξ〉 = −〈f ′′0 (z0eiσ)φ∗µ,ξ, φ

∗µ,ξ〉,

where φ∗µ,ξ(x−ξµ ) = µN/2−1φµ,ξ(x) and f ′′0 (z0e

iσ)φ∗µ,ξ = kµ,ξ, where

kµ,ξ(y) =2

iµeiσ∇yz0(y) ·A(µy + ξ) +

µ2

ieiσ divy A(µy + ξ)z0(y).

As µ→ 0+, we have kµ,ξ → kξ, where

kξ(y) :=2

ieiσ∇yz0(y) ·A(ξ).


Let us define ψξ(x) = limµ→0+Lz0kµ,ξ

µ = limµ→0+φ∗µ,ξ

µ . We have that

f ′′0 (z0eiσ)ψξ =

2

ieiσ∇xz0(y) ·A(ξ). (1.6.22)

Setting gξ(x) = e−iσψξ(x), we have that for any φ ∈ D1,2(RN ,R)

〈f ′′0 (z0eiσ)eiσgξ, e

iσφ〉 = Re

∫

RN

2

ieiσ∇yz0(y) ·A(ξ)e−iσφdy = 0.

This means that for any φ ∈ D1,2(RN ,R)

0 = 〈f ′′0 (z0eiσ)eiσgξ, e

iσφ〉

= Re

∫

RN

∇(eiσgξ) · ∇(eiσφ) − Re

∫

RN

|z0|2∗−2eiσgξeiσφ

− Re(2∗ − 2)

∫

RN

|z0|2∗−4 Re(eiσz0e

iσgξ)eiσz0eiσφ

= Re

∫

RN

∇(gξ) · ∇φ− Re

∫

RN

|z0|2∗−2gξφ

− Re(2∗ − 2)

∫

RN

|z0|2∗−4 Re(z0gξ)z0φ

=

∫

RN

∇(Re gξ) · ∇φ−

∫

RN

|z0|2∗−2 Re gξφ

− (2∗ − 2)

∫

RN

|z0|2∗−4 Re gξz0

2φ

= 〈f ′′0 (z0)Re gξ, φ〉.

It follows that Re gξ = 0 as φµ,ξ ∈(Teiσzµ,ξ

Z)⊥

. Therefore ψξ(x) =

ieiσrξ(x) with rξ ∈ D1,2(RN ,R). Now we test (1.6.22) against functions of thetype ieiσω(x), ω ∈ D1,2(RN ,R). It follows that:

Re

∫

RN

2

ieiσ∇xz0(x) ·A(ξ)ieiσw =

⟨f ′′0 (z0e

iσ)ψξ, ieiσw⟩

= Re

∫

RN

∇rξ · ∇w − Re

∫

RN

|z0|2∗−2rξw

−Re(2∗ − 2)

∫

RN

|z0|2∗−4 Re(iz0rξ)z0iw

or equivalently

Re

∫

RN

∇rξ · ∇w − Re

∫

RN

|z0|2∗−2rξw = −Re

∫

RN

2∇xz0(x) ·A(ξ)w.


We deduce that rξ satisfies the equation

−∆rξ(x) − |z0|2∗−2rξ(x) = −2∇z0 ·A(ξ). (1.6.23)

We notice that the function u(x) = z0(x)A(ξ)·x solves the equation (1.6.23), as∆u = ∆z0A(ξ)·x+z0∆(A(ξ)·x)+2∇z0 ·∇(A(ξ)·x) = ∆z0A(ξ)·x+2∇z0 ·A(ξ).

Since iz0(x)(A(ξ)|x)eiσ belongs to(Teiσz0Z

)⊥, we deduce that ψξ(x) =

ieiσz0(x)A(ξ) · x and thus

limµ→ 0+

1

2

〈G′1(zµ,ξ), φµ,ξ〉

µ2= −Re

∫

RN

1

ieiσ∇yz0(y) ·A(ξ) ieiσz0A(ξ) · y dy

=

∫

RN

∇yz0(y) ·A(ξ)z0 A(ξ) · y dy.

Since we have∫

RN

∇yz0(y) ·A(ξ)z0A(ξ) · y dy

= −

∫

RN

∇yz0(y) ·A(ξ)z0A(ξ) · y dy −

∫

RN

|A(ξ)|2z20 dy,

we conclude that

limµ→ 0+

1

2

〈G′1(zµ,ξ), φµ,ξ〉

µ2=

∫

RN

∇yz0(y) ·A(ξ)z0A(ξ) · y dy

= −1

2

∫

RN

|A(ξ)|2z20 dx. (1.6.24)

Therefore we have that

limµ→ 0+

Γ(µ, ξ)

µ2= lim

µ→ 0+

1

µ2(G2(µ, ξ) +

1

2

⟨G′

1(zµ,ξ), φµ,ξ⟩) =

1

2V (ξ)

∫

RN

|z0|2.

Remark 1.6.12. The presence of a non-trivial potential V is crucial in theprevious Proposition. Otherwise, from (1.6.21) and (1.6.24) we would simply

get that limµ→0+Γ(µ,ξ)µ2 = 0, and Γ might still be a constant function. Hence

V is in competition with A. It would be interesting to investigate the case inwhich V = 0 identically. We conjecture that some additional assumptions onthe shape of A should be made.

1.7. Proof of the main result 45

1.7 Proof of the main result

In this Section, we can finally apply the abstract Theorem 1.2.3 to find thecritical point of fε and prove our main existence result for equation (1.3.3).According to Remark 1.3.1, we will use the term solution rather than the moreprecise S1–orbit of solutions.

Theorem 1.7.1. Retain assumptions (N), (A1–2), (V). Assume that V (ξ) 6=0 for some ξ ∈ R

N . Then, there exists ε0 > 0 such that for all ε ∈ (0, ε0) equa-tion (1.3.3) possesses at least one solution uε ∈ E. If V is a changing signfunction, then there exists two solutions of equation (1.3.3).

Proof. Under our assumptions, the Melnikov function Γ, extended across thehyperplane µ = 0 by reflection, is not constant and possesses at least acritical point (either a minimum or a maximum point). We can thereforeinvoke Theorem 1.2.3 to conclude that there exists at least one solution uε to(1.3.3), provided ε is small enough. If there exist points ξi ∈ R

N , i = 1, 2,such that V (ξ1)V (ξ2) < 0, then it follows from the previous Proposition thatΓ must change sign near µ = 0. In particular, it must have both a minimumand a maximum. Hence there exist two different solutions to (1.3.3).

Remark 1.7.2. Consider equation (1.3.3). It is clear that our main theoremstill applies for any α ∈ [1, 2). Indeed, in the expansion (1.2.5), the lowestorder term in ε is

εα∫

RN

V z2 dx,

and consequently the magnetic potential A no longer affects the finite dimen-sional reduction. In some sense, we have treated with the more all the detailsthe “worst” situation in the range 1 ≤ α ≤ 2.


1.8 Regularity results

In the following Sections we recall certain regularity properties of solutionsfor a rather general class of Schrodinger equations with electromagnetic fieldestablished in the paper [36] by Chabrowski and Szulkin. Then, we provethat these results apply to solutions to (1.3.3) . Precisely, we consider thesemilinear Schrodinger equation

−∆Au+ V (x)u = Q(x)|u|2∗−2u, u ∈ H1

A,V (RN ), (1.8.1)

where −∆A = (−i∇+A)2, u : RN → C, N ≥ 3, 2∗ := 2N/(N−2) is the critical

Sobolev exponent. The coefficient V is the scalar (or electric) potential andA = (A1, ....., AN ) : R

N → RN the vector (or magnetic) potential. Throughout

these Sections we assume that A ∈ L2loc(R

N ), V ∈ L1loc(R

N ) and V − ∈LN/2(RN ). Here V − is the negative part of V , that is V −(x) = max(−V (x), 0).It is assumed that the coefficient Q is positive, continuous and bounded onRN . Further assumptions on Q will be formulated later.

We now define some Sobolev spaces. By D1,2A (RN ) we denote the Sobolev

space defined by

D1,2A (RN ) =

u : u ∈ L2∗(RN ),∇Au ∈ L2(RN )

,

where ∇A = (∇ + iA). The space D1,2A (RN ) is a Hilbert space with the inner

product ∫

RN

∇Au∇Av dx.

It is known that the space C∞0 (RN ) is dense in D1,2

A (RN ) (see [55]). Equiva-

lently D1,2A (RN ) can be defined as the closure of C∞

0 (RN ) with respect to thenorm

||u||2D1,2

A

=

∫

RN

|∇Au|2 dx.

ByH1A,V +(RN ) we denote the Sobolev space obtained as the closure of C∞

0 (RN )with respect to the norm

||u||2H1A,V +

=

∫

RN

(|∇Au|2 + V +(x)|u|2) dx,

where V +(x) = max(V (x), 0). H1A,V +(RN ) is a Hilbert space with the inner

product ∫

RN

(∇Au∇Av + V +(x)uv) dx.

1.9. The regularity of solutions involving the operator ∆A 47

Obviously, we have a continuous embedding H1A,V +(RN ) ⊂ D1,2

A (RN ); hence

in particular, |u| ∈ L2∗(RN ) whenever u ∈ H1A,V +(RN ).

We shall frequently use in these Sections the diamagnetic inequality (see [75])

|∇|u|| ≤ |∇Au| a.e. in RN . (1.8.2)

This inequality implies that if u ∈ H1A,V +(RN ), then |u| ∈ D1,2(RN ), where

D1,2(RN ) is the usual Sobolev space of real valued functions defined by

D1,2(RN ) =u : u ∈ L2∗(RN ),∇u ∈ L2(RN )

.

Solutions of (1.8.1) will be sought in the Sobolev space H1A,V +(RN ) as critical

points of the functional

J(u) =1

2

∫

RN

(|∇Au|2 + V (x)|u|2) dx−

1

2∗

∫

RN

Q(x)|u|2∗dx.

It is easy to see that J is a C1-functional on H1A,V +(RN ). For the Palais-Smale

condition for the variational functional J and the existence results for (1.8.1)we refer to Sections 3 and 4 in [36] where the authors firstly solve a weightedlinear eigenvalue problem for the operator −∆A + V +. If the first eigenvalueµ1 > 1, then a solution is obtained through a constrained minimization. Thissituation has already been envisaged in the paper [14]. If µ1 ≤ 1, they employa topological linking argument.Now, in Section 1.9 we devote to the regularity properties of solutions of(1.8.1). We show that solutions in H1

A,V +(RN ) are bounded and decay to 0 atinfinity.

1.9 The regularity of solutions involving the opera-

tor ∆A.

Let V be a nonnegative function in L1loc(R

N ). We commence by establishingthe integrability properties of solutions of the equation

−∆Au+ V (x)u = g(x)u in RN . (1.9.1)

It is assumed that g : RN → R is a measurable function satisfying

|g(x)| ≤ a+ b(x) on RN ,


where a ≥ 0 is a constant and b is a nonnegative function in LN/2(RN ). Let

φ(x) = η(x)2u(x)min(|u(x)|β−1, L),

where β > 1 and L > 0 are constants, u ∈ H1A,V (RN ) and η is a C1-real valued

function which is bounded together with its derivatives.In what follows, χΩ denotes the characteristic function of the set Ω. Bystraightforward computations we have

∇Aφ = 2η∇ηumin(|u|β−1, L) + η2∇Aumin(|u|β−1, L)

+ (β − 1)η2u|u|β−2∇|u|χ|u|β−1<L

and

∇Au∇Aφ = |∇Au|2η2 min(|u|β−1, L) + 2η∇ηumin(|u|β−1, L)∇Au

+ (β − 1)η2u|u|β−2∇|u|χ|u|β−1<L∇Au.

We now observe that

Re(u∇Au) = Re(∇u+ iAu)u = Re(u∇u) = |u|Re

(u

|u|∇u

)= |u|∇|u|.

Taking the real part of ∇Au∇Aφ we obtain the following inequality:

Re(∇Au∇Aφ) = |∇Au|2η2 min(|u|β−1, L)

+2η∇η∇|u||u|min(|u|β−1, L) + (β − 1)η2|u|β−1|∇|u||2χ|u|β−1<L

≥ |∇Au|2η2 min(|u|β−1, L) + 2η∇η∇|u||u|min(|u|β−1, L). (1.9.2)

Lemma 1.9.1. Solutions of equation (1.9.1) in H1A,V (RN ) belong to Lp(RN )

for every p ∈ [2∗,∞).

Proof. We adapt to our case an argument which may be found e.g. in [98](Appendix B). We test equation (1.9.1) with φ = umin(|u|β−1, L). It thenfollows from inequality (1.9.2), with η = 1, that for every constant K > 0 we


have∫

RN

|∇Au|2 min(|u|β−1, L) dx

≤ a

∫

RN

|u|2 min(|u|β−1, L) dx+K

∫

b(x)≤K|u|2 min(|u|β−1, L) dx

+

(∫

b(x)>Kb(x)N/2 dx

)2/N

·

(∫

RN

(|u|min(|u|(β−1)/2, L1/2))2∗dx

)(N−2)/N

≤ (a+K)

∫

RN

|u|2 min(|u|β−1, L) dx+

(∫

b(x)>Kb(x)N/2 dx

)2/N

·

(∫

RN

(|u|min(|u|(β−1)/2, L1/2))2∗dx

)(N−2)/N

. (1.9.3)

On the other hand, by the diamagnetic inequality we have∫

RN

|∇|u||2 min(|u|β−1, L) dx ≤

∫

RN

|∇Au|2 min(|u|β−1, L) dx. (1.9.4)

We also have∫

RN

|∇(|u|min(|u|(β−1)/2, L1/2))|2 dx

≤ 2

∫

RN

|∇|u||2 min(|u|β−1, L) dx+(β − 1)2

2

∫

RN

|∇|u||2|u|β−1χ|u|β−1<L dx

≤

(2 +

(β − 1)2

2

)∫

RN

|∇|u||2 min(|u|β−1, L) dx. (1.9.5)

Combining (1.9.3)-(1.9.5) we obtain∫

RN

|∇(|u|min(|u|(β−1)/2, L1/2))|2 dx

≤ (a+K)

(2 +

(β − 1)2

2

)∫

RN

|u|2 min(|u|β−1, L) dx

+

(2 +

(β − 1)2

2

)(∫

b(x)>Kb(x)N/2 dx

)2/N

·

(∫

RN

(|u|min(|u|(β−1)/2, L1/2))2∗dx

)(N−2)/N

.


Since∫b(x)>K b(x)

N/2 dx → 0 as K → ∞, taking K sufficiently large andapplying the Sobolev inequality to the left-hand side above, we obtain

(∫

RN

(|u|min(|u|(β−1)/2, L1/2))2∗dx

)2/2∗

≤ C1(K,β)

∫

RN

|u|2 min(|u|β−1, L) dx (1.9.6)

for some constant C1(K,β) > 0. We now set β + 1 = 2∗. Letting L → ∞ wederive from the above inequality that

(∫

RN

|u|(2∗N)/(N−2) dx

)2/2∗

≤ C1(K, 2∗)

∫

RN

|u|2∗dx

and thus u ∈ L(2∗N)/(N−2)(RN ). A standard application of a boot-strap argu-ment to (1.9.6) completes the proof.

Proposition 1.9.2. If u ∈ H1A,V (RN ) is a solution of (1.9.1), then u ∈

L∞(RN ) and lim|x|→∞ u(x) = 0.

Proof. We follow some ideas from the proof of Theorem 8.17 in [60] (in partic-ular, we use Moser’s iteration technique). Let η be a C1-function in R

N witha compact support. Testing (1.9.1) with φ = η2umin(|u|β−1, L) and usinginequality (1.9.2) we obtain the estimate

∫

RN

|∇Au|2η2 min(|u|β−1, L) dx+ 2

∫

RN

η∇η∇|u||u|min(|u|β−1, L) dx

≤

∫

RN

b|u|2η2 min(|u|β−1, L) dx+ a

∫

RN

|u|2η2 min(|u|β−1, L) dx.

Hence by the diamagnetic inequality and since

1

2η2|∇|u||2 − 2|u|2|∇η|2 ≤ η2|∇|u||2 + 2η|u|∇|u|∇η,

we get

1

2

∫

RN

|∇|u||2η2 min(|u|β−1, L) dx ≤

∫

RN

b|u|2η2 min(|u|β−1, L) dx

+2

∫

RN

|∇η|2|u|2 min(|u|β−1, L) dx+ a

∫

RN

|u|2η2 min(|u|β−1, L) dx.


Letting L→ ∞ we obtain

1

2

∫

RN

|∇|u||2η2|u|β−1 dx ≤

∫

RN

b|u|β+1η2 dx

+2

∫

RN

|∇η|2|u|β+1 dx+ a

∫

RN

|u|β+1η2 dx.

Substituting w = |u|(β+1)/2 in this inequality, we obtain

2

(β + 1)2

∫

RN

|∇w|2η2 dx ≤

∫

RN

bw2η2 dx

+2

∫

RN

|∇η|2w2 dx+ a

∫

RN

w2η2 dx. (1.9.7)

We now observe that

∫

RN

|∇(wη)|2 dx ≤ 2

∫

RN

|∇w|2η2 dx+ 2

∫

RN

|∇η|2w2 dx,

which combined with (1.9.7) gives

∫

RN

|∇(wη)|2 dx ≤ (β + 1)2∫

RN

bw2η2 dx

+2((β + 1)2 + 1)

∫

RN

|∇η|2w2 dx+ a(β + 1)2∫

RN

w2η2 dx.

It then follows from the Holder and Sobolev inequalities that

S

(∫

RN

(wη)2∗dx

)(N−2)/N

≤ (β + 1)2(∫

RN

bN/2 dx

)2/N (∫

RN

(wη)2∗dx

)(N−2)/N

+2((β + 1)2 + 1)

∫

RN

|∇η|2w2 dx

+a(β + 1)2∫

RN

w2η2 dx, (1.9.8)

where

S = inf

∫

RN

|∇u|2 dx : u ∈ C∞0 (RN ),

∫

RN

|u|2∗dx = 1


is the Sobolev constant. To proceed further we choose R > 0 so that

(β + 1)2

(∫

|x|>RbN/2 dx

)2/N

≤S

2.

Assuming that supp η ⊂ (|x| > R) we derive from (1.9.8) that

S

(∫

RN

(wη)2∗dx

)(N−2)/N

≤ 4((β + 1)2 + 1)

∫

RN

|∇η|2w2 dx

+ 2a(β + 1)2∫

RN

w2η2 dx. (1.9.9)

We now make a more specific choice of η: η ∈ C1(RN , [0, 1]), η(x) = 1 inB(x0, r1), η(x) = 0 in R

N \ B(x0, r2), |∇η(x)| ≤ 2/(r2 − r1) in RN , 1 ≤ r1 <

r2 ≤ 2. It is also assumed that B(x0, r2) ⊂ (|x| > R). It then follows from(1.9.9) that

(∫

B(x0,r1)w2∗ dx

)1/2∗

≤A(β + 1)

r2 − r1

(∫

B(x0,r2)w2 dx

)1/2

,

where A is an absolute constant. Setting γ = β + 1 = 2∗, χ = N/(N − 2) weget

(∫

B(x0,r1)|u|γχ dx

)1/γχ

≤

(Aγ

r2 − r1

)2/γ(∫

B(x0,r2)|u|γ dx

)1/γ

.

To iterate this inequality (which holds for any γ ≥ 2∗), we take sm = 1+2−m,r1 = sm, r2 = sm−1 and replace γ = 2∗ by γχm−1, m = 1, 2, ...... Then we get

(∫

B(x0,sm)|u|χ

mγ dx

)1/(γχm)

≤

(Aγχm−1

sm−1 − sm

)2/(γχm−1)(∫

B(x0,sm−1)|u|χ

m−1γ dx

)1/(γχm−1)

= (Aγ)2/(γχm−1)2(2m)/(γχm−1)χ2(m−1)/(γχm−1)

·

(∫

B(x0,sm−1)|u|χ

m−1γ dx

)1/(γχm−1)

,


and by induction,

(∫

B(x0,sm)|u|χ

mγ dx

)1/(γχm)

≤ (Aγ)(2/γ)∑m−1

j=0 (1/χj)2(2/γ)∑m−1

j=0 ((j+1)/χj)χ(2/γ)∑m−1

j=0 (j/χj)

·

(∫

B(x0,s0)|u|γ dx

)1/γ

,

for each m > 1. Since s0 = 2 and sm → 1, we deduce the following estimateby letting m→ ∞: there exist constants R > 0 and C > 0 such that for everyB(x0, 2) ⊂ (|x| > R) we have

supB(x0,1) |u(x)| ≤ C

(∫

B(x0,2)|u|γ dx

)1/γ

.

This inequality yields lim|x|→∞ |u(x)| = 0. To prove the boundedness of u inthe ball B(0, R) we fix x ∈ B(0, R), choose r > 0 so that

(β + 1)2

(∫

B(x,r)bN/2 dx

)2/N

≤S

2,

and then let η have support in B(x, r). We now repeat the previous argumentwith a suitable rescaling in the ball B(x, r) to obtain the boundedness of u inB(x, r/2). By a standard compactness argument we show that u is boundedin B(0, R). This combined with the first part of the proof shows that u ∈ L∞.

We now observe that any solution u ∈ H1A,V +(RN ) of the equation

−∆Au+ V (x)u = f(x, |u|)u, (1.9.10)

where |f(x, |u|)| ≤ c(1 + |u|2∗−2), satisfies

−∆Au+ V +(x)u = (V −(x) + f(x, |u|))u ≡ g(x)u. (1.9.11)

Since |g(x)| ≤ c + (V −(x) + c|u(x)|2∗−2) and V − ∈ LN/2(RN ), |u|2

∗−2 ∈L2∗/(2∗−2)(RN ) = LN/2(RN ), we can state the following result:


Corollary 1.9.3. Let u ∈ H1A,V +(RN ), N ≥ 3, be a solution of (1.9.10).

Then u ∈ L∞(RN ) and

lim|x|→∞

u(x) = 0

(in the sense that limR→∞ ||u||L∞(RN\B(0,R)) = 0).

Remark 1.9.4. Let N = 2. If u ∈ H1A,V +(R2) ∩ L2(R2), then u ∈ Lp(R2)

for all p ∈ [2,∞] by the diamagnetic inequality and the Sobolev embeddingtheorem. Suppose g(x) in (1.9.1) is such that b ∈ Lq(R2) for some q ∈ (1, 2)and u ∈ H1

A,V +(R2) ∩ L2(R2) is a solution of (1.9.1). Then the conclusion ofProposition 1.9.2 remains valid. Indeed, the argument employed there appliesexcept that the L2∗-norm in (1.9.8) should be replaced by the Lq

′-norm, where

q′ = q/(q − 1), and one needs to take γ = β + 1 = q′, χ = q′/2. Alsothe conclusion of Corollary 1.9.3 remains valid if u ∈ H1

A,V +(R2) ∩ L2(R2),

V − ∈ Lq(R2) and |f(x, |u|)| ≤ c(1 + |u|r) for some q ∈ (1, 2) and r > 0.

In particular, Corollary 1.9.3 (or Remark 1.9.4 ifN = 2) applies to all solutionsfound in [14], [55] as well as the solutions found in Theorems 4.1, 4.2 andCorollary 4.3 in [36].

Remark 1.9.5. If V is a nonnegative function, i.e. V − = 0, by hypotheses(A1) and (V) in Section 1.5, it follows that every u ∈ D1,2(RN ) belongs toH1A,V +(RN ). Then the conclusion of Corollary 1.9.3 applies to all the solutions

to (1.3.3) found in the previous Sections via perturbation methods.

As an application of Corollary 1.9.3 and Remark 1.9.4 we establish anexponential decay of solutions of (1.9.10). However, we need additional as-sumptions on V and f .

Proposition 1.9.6. Suppose that f ≥ 0, f(x, 0) = 0, V + ∈ Lploc(RN ) and

V − ∈ Lp(RN ) for some p > N/2. Moreover, assume that there exist constantsa > 0 and R > 0 such that V (x) ≥ a for |x| ≥ R. If u ∈ H1

A,V +(RN ) is a

solution of (1.9.10), then

|u(x)| ≤ Ce−α|x| a.e. in RN ,

where α2 = a/2.

Proof. Since V ≥ a for |x| ≥ R, it is easy to see that u ∈ L2(RN ), and henceu ∈ Lq(RN ) for all 2 ≤ q ≤ ∞ according to Corollary 1.9.3 (or Remark 1.9.4).Therefore there exists a unique solution v ∈ H1(RN ) of the equation


−∆v + V +(x)v = (V −(x) + f(x, |u|))|u|,

and by standard regularity theory and the maximum principle v is continuousand ≥ 0. Moreover, it follows from (1.9.11) and Theorem B.13.2 in [96] that|u| ≤ v a.e. (more precisely, one obtains this inequality by integrating (B41)of [96] from t = 0 to t = ∞; the hypothesis that p > N/2 is used in order tohave v continuous and V + ∈ K loc

N , V − ∈ KN in the notation of [96]). Now itremains to establish the exponential decay of v. We follow the argument usedin Proposition 4.4 from [99]. Since v satisfies

−∆v + V +(x)v ≤ (V −(x) + f(x, |u|))v in RN ,

we have

−∆v ≤ (−V (x) + f(x, |u|))v ≤ −a

2v for |x| ≥ R

by taking R larger if necessary. Let

W (x) = Me−α(|x|−R)

andΩ(L) = x : R < |x| < L and v(x) > W (x) ,

where a constant M > 0 is chosen so that v(x) ≤ W (x) for |x| = R. Ifα2 = a/2, we get

∆(W − v) =

(α2 −

α(N − 1)

|x|

)W − ∆v ≤ α2(W − v) ≤ 0

on Ω(L). By the maximum principle

W (x) − v(x) ≥ minx∈∂Ω(L)(W − v) ≥ min(0,min|x|=L(W − v)).

Since lim|x|→∞ v(x) = lim|x|→∞W (x) = 0, letting L→ ∞, we deduce that

v(x) ≤W (x) = Me−α(|x|−R) for |x| ≥ R.

Chapter 2


equations with singular

electric potential

2.1 A review on Ljusternik-Schnirelman theory

The Ljusternik-Schnirelman Theory is, jointly with the Morse Theory, one ofthe most classical and powerful tool in Critical Point Theory. In this Chapterwe attempt to highlight the main ideas of LS minimax method which has beendeveloped by Ljusternik and Schnirelman in [78] and extended in differentcases in [34, 69, 85, 92, 101]. The key-idea is to construct critical values of aC1-functional on a compact manifold and find a related multiplicity result viaLS category. For details, we refer to Ambrosetti [3], Mawhin-Willem [81] andStruwe [98].Let X be a topological space and A ⊂ X, A 6= ∅. A map ϕ ∈ C(A,X) is adeformation if there is a homotopy h ∈ C([0, 1] ×A,X) such that

h(0, ·) = ϕ, h(1, ·) = identity.

A is contractible (to a point u0) in X if there is a deformation ϕ ∈ C(A,X)such that ϕ(u) = u0.The category of A relative to X , cat(A; X), is the smallest integer k such that

A ⊂ A1 ∪ ......... ∪Ak

58 Chapter 2. Magnetic Schrodinger equations with singular electric potential

with Ai closed and contractible in X, for each i = 1, ....., k. If there are no suchintegers, we set cat(A;X) = +∞. We set also cat(∅;X) = 0, and abbreviatecat(X) for cat(X;X).The main properties of the category are collected in the following

Lemma 2.1.1. Let A,B ⊂ X.

(i) (Monotonicity): if A ⊆ B then cat(A;X) ≤ cat(B;X);

(ii) (Subadditivity): cat(A ∪B;X) ≤ cat(A;X) + cat(B;X);

(iii) (Supervariance): If ϕ ∈ C(A,X) is a deformation and A is closed, then

cat(ϕ(A);X) ≥ cat(A;X);

(iv) (Continuity): let X be an ANR (Absolute Neighbourhood Retract)1 andK ⊂ X be compact. Then cat(K;X) < +∞ and there exists a neigh-bourhood U of K such that cat(U ;X) = cat(K;X);

(v) (Multiplicity): if A is closed and cat(A;X) = n ∈ N, n 6= 0, then

#(A) ≥ cat(A;X) = n.

Examples 2.1.2. (i) Let SN−1 =x ∈ R

N : |x| = 1. One has:

cat(SN−1) = 2.

(ii) Let T k = S1 × S1 × ......× S1 (k times) denote the k-dimensional torus.There results cat(T k) = k + 1.

(iii) Let us consider the representation of Z2 over RN

T (0) = id, T (1) = −id,

and let PN = SN−1/Z2 denote the corresponding projective space. Asa consequence of the Borsuk Antipodensatz ([81], Chapter 5) one provesthat cat(PN ) = N . In addition, if E is a separable, infinite dimensionalHilbert space and S∞ = u ∈ E : ||u|| = 1, letting P∞ = S∞/Z2, thereresults cat(P∞) = +∞.

1A metric space X is an ANR if, for every metric space E, every closed subset F of E

and every continuous map f : F → X, there exists a continuous extension of f defined on aneighbourhood of F in E.

2.2. Minimax theorem 59

2.2 Minimax theorem

The category can be employed to find critical levels of min-max type.For all k ≤ cat(X), we consider the class Ak of all subsets A ⊂ X such thatcat(A;X) ≥ k and define

ck = infA∈Ak

[sup f(u) : u ∈ A].

Note that since Ak ⊃ Ak+1 then

c1 ≤ c2 ≤ ..... ≤ ck ≤ ck+1 ≤ .....

In order to show that ck’s are critical levels, we suppose that

(A0) X = M is a complete, C1 Hilbert manifold and f ∈ C1(M,R).

In addition, the following compactness condition introduced by Palais andSmale [84] is in order. We say that (M,f) satisfies (PS), or simply that (PS)holds, if any sequence un ⊂M such that

|f(un)| ≤ const. (2.2.1)

f ′M (un) → 0, (2.2.2)

has a converging subsequence.Any sequence satisfying (2.2.1) and (2.2.2) will be called a PS-sequence.

Theorem 2.2.1. Let (M,f) satisfy (A0), (PS) and let f be bounded frombelow on M :

f(u) ≥ a0, ∀u ∈M.

Then:

(i) each ck < +∞ is a critical level for f on M ;

(ii) if c := ck = ck+1 = .... = ck+m and Kc = u ∈ K : f(u) = c whereK = u ∈M : f ′M (u) = 0, then cat(Kc;M) ≥ m+ 1;

(iii) if ck = +∞ for some k, then supf(u) : u ∈ K = +∞.

In particular, f has at least cat(M) critical points on M (Lemma 2.1.1 (v)).


Remark 2.2.2. If E is finite dimensional and M is compact, Theorem 2.2.1goes back to Ljusternik-Schnirelman [78]. For the extension to infinite dimen-sion (under the assumption that both f and M are C2) see, for example, [69]and [92]. Palais [85] handled the case of Finsler manifolds M modeled ona Banach space (see also Browder [34]) and C1 functionals. Finally, Szulkin[101] has weakened the regularity assumptions on M , showing that C1 suffices.

In order to highlight the role of (PS), let us outline the proof of (ii). Firstof all, one uses (PS) to deduce

Lemma 2.2.3. Let c ∈ R and suppose (M,f) verifies (PS). Then

(i) Kc is compact;

(ii) for all ε > 0 and any neighbourhood U of Kc there exists α > 0 suchthat ||f ′M (u)|| ≥ α > 0 for all u ∈ f c+εc−ε \ U .

Using Lemma 2.2.3 (ii) one proves the following

Lemma 2.2.4 (Deformation Lemma). Let f be as in Theorem 2.2.1.

(i) If Kc = ∅, ∀c ∈ [a, b], then f b can be deformed in fa.

(ii) Given c ∈ R, ε ∈ (0, 12 ] and any neighbourhood U of Kc there exist

δ ∈ (0, ε) and σ ∈ C([0, 1] ×M,M) such that, for all δ < d < 12 , there

resultsσ(0, u) = u, ∀u ∈M (2.2.3)

σ(t, u) = u, ∀ 0 ≤ t ≤ 1, ∀u /∈ f c+dc−d (2.2.4)

f(σ(t, u)) ≤ f(u), ∀ 0 ≤ t ≤ 1, ∀u ∈ M (2.2.5)

f(σ(1, u)) < c− δ, ∀u ∈ f c+δ \ U. (2.2.6)

Roughly, if f is C2, σ is found by using the flow generated by a Cauchyproblem like

σ′ = X(σ)

σ(0) = u

where σ′ = dσ/dt and X is the locally Lipschitzian vector field such thatX = −f ′M in the strip f c+δc−δ and X = 0 in the complement of the strip f c+dc−d . Iff is merely C1 one uses the so called Pseudo-gradient Vector Fields introducedin [85].It is worth pointing out that (2.2.6) follows from Lemma 2.2.3 (ii).

2.3. A multiplicity result 61

Finally, the Deformation Lemma is used to define a deformation ϕ := σ(1, ·)with the property

ϕ(A \ U) ⊂ f c−δ

for all A ⊂ f c+δ (δ > 0 small enough, U neighbourhood of Kc).

We are now in position to prove the claim (ii) of Theorem 2.2.1.Suppose, by contradiction, that

cat(Kc;M) ≤ m.

Since Kc is compact (Lemma 2.2.3 (i)) we can use Lemma 2.1.1 (iv) to find aneighbourhood U of Kc such that

cat(U ;M) = cat(Kc;M) ≤ m.

Using the definition of c = ck+m, there is A ∈ Ak+m such that A ⊂ f c+δ. LetA′ = A \ U . From Lemma 2.1.1 (ii) it follows that cat(A′;M) ≥ cat(A;M) −cat(U ;M) ≥ k +m−m = k, namely A′ ∈ Ak. Then, for ϕ(A′) one has:

ϕ(A′) ∈ Ak (Lemma 2.1.1 (iii))

ϕ(A′) ⊂ f c−δ (from (2.2.3)).

These two relationships are in contradiction with the definition of c = ck.

2.3 A multiplicity result

Among the possible applications of Theorem 2.2.1 and its variants, let usrecall a multiplicity result for critical points involving Ljusternik-Schnirelmancategory, which we shall apply in proving Theorem 2.4.2 (for the proof, e.g.see [81, 98]).

Theorem 2.3.1. Let N be a C1,1 complete Riemannian manifold (modelledon a Hilbert space) and assume h ∈ C1(N ,R) bounded from below. Let −∞ <infN h < a < b < +∞. Suppose that h satisfies Palais-Smale condition on thesublevel u ∈ N : h(u) ≤ b and that a is not a critical level for h. Then

# u ∈ ha : ∇h(u) = 0 ≥ catha(ha),

where ha ≡ u ∈ N : h(u) ≤ a.


With a view to apply Theorem 2.3.1, the following abstract Lemma providea very useful tool in that it relates the topology of some sublevel of a functionalto the topology of some subset of the space R

N . For the proof, an easyapplication of the definitions of category and of homotopic equivalence betweenmaps, we refer to [20, 21, 22, 48].

Lemma 2.3.2. Let H, Ω+ and Ω− be closed sets with Ω− ⊂ Ω+; let β : H →Ω+, ψ : Ω− → H be two continuous maps such that β ψ is homotopicallyequivalent to the embedding j : Ω− → Ω+. Then, catH(H) ≥ catΩ+(Ω−).

2.4 A multiplicity result for singular NLS equations

with magnetic potentials

Let us consider the nonlinear Schrodinger equation with an external magneticfield

i~∂ψ

∂t=

(~

i∇−A(x)

)2

ψ + U(x)ψ − |ψ|p−2ψ, x ∈ RN , (2.4.1)

with 2 < p < 2∗. The search of standing waves solutions ψ(x, t) = e−iE~tu(x),

E ∈ R, to (2.4.1) when ~ is sufficiently small, leads to seek for solutionsu : R

N → C for the complex equation in RN :

(~

i∇−A(x)

)2

u+ (U(x) − E)u = |u|p−2u, x ∈ RN . (2.4.2)

For simplicity, let V (x) = U(x) − E and assume that V is strictly positive.The transition from Quantum Mechanics to Classical Mechanics can be for-mally described by letting ~ → 0 and thus the existence of solutions for ~

small has physical interest. Standing waves for ~ small are usually referred assemiclassical bound states.The search of semiclassical standing waves solutions to (2.4.1) in the caseA = 0 (i.e. no magnetic vector potential) and a nonlinear subcritical termhas been studied in several papers (see [4, 9, 39, 40, 41, 42, 51, 52, 53, 57, 62,74, 82, 89, 106]). In particular, Wang [105] addressed concentration of boundstates, proving, for ground states, that any sequence of ground states containsa subsequence which concentrates at a global minimum of V (x) as ~ → 0. Insuch a way, he established a link between solutions to (2.4.2) in the case A = 0and stationary points of V and gave rise to the study of multiplicity resultsfor such solutions.The first existence result of standing wave solutions to (2.4.1) is due to M.Esteban and P.L. Lions [55] in the case N = 2 and 3, V = 1 identically, ε > 0fixed, by a constrained minimization. These problems which are a priori notcompact are solved by using the concentration-compactness method. Morerecently, Kurata [70] has proved the existence of a least energy solution of(2.4.1) for any fixed ~ > 0, under some assumptions linking the magnetic fieldB and the electric potential V (see also [94]).A first multiplicity result for standing wave solutions to (2.4.2) (A 6= 0), hasbeen proved by Cingolani in [45], using topological arguments that allow to


relate the number of standing wave solutions to (2.4.2) to the topology of theset of global minima of V . Indeed, the sequence of the modulus of such solu-tions exhibits a concentration behavior at any global minimum point of V for~ small. The magnetic field only influences the phase factor of the solutionsas ~ is small. This result covers the case of magnetic potentials having poly-nomial growths, having special physical interest, but the used approach worksonly near global minima of V . In the case A = 0, we refer to [39] and [42] forcompeting potential functions.Concerning bounded vector potentials, existence and concentration phenom-ena at any non-degenerate critical point, not necessarily a minimum, of V , as~ → 0, (i.e. V has a manifold of stationary points), are proved by Cingolaniand Secchi in [44] using a perturbation approach contained in the paper [9] byAmbrosetti, Malchiodi, Secchi (see also [4, 6]).

In [47], using a penalization procedure (see [53]), Cingolani and Secchiextend the result in [44] in the case of a vector potential A possibly unboundedand magnetic fields constant. A result concerning existence of semiclassicalmulti-peak solutions for (2.4.2) for bounded vector potentials is contained inthe recent paper [35] by Cao and Tang, and for unbounded domains in [19]by Bartsch, Dancer and Peng. Concerning other papers on this topic, wemention a recent work [93] by Secchi and Squassina in which the authors haveestablished necessary conditions for a sequence of standing wave solutions to(2.4.2) to concentrate, in different senses, around a given point. Dealing withperiodic potentials, we mention the paper [14] by Arioli and Szulkin whereexistence of infinitely many solutions of (2.4.2) is proved for each ~ fixed.Our goal is to show that a result in the spirit of the paper by Cingolani [45]holds if the potential in (2.4.2) is perturbed by adding a negative potentialwhich may be singular, so that the resulting potential may be unboundedbelow.Precisely, we consider the potential

Vε(x) = V (x) − ε(~)W (x)

where ε : [0,+∞) → [0,+∞) and W : RN → [0,+∞) is a measurable function

such that, for some α1 > 0 and α2 ≥ 0, the inequality∫

RN

W (x)|v|2 ≤ α1||∇|v|||22 + α2||v||22 (2.4.3)

holds for any v such that |v| ∈ H1(RN ,R). Such a condition has been in-troduced by Lazzo in [73] to obtain multiple real-valued solutions in the

2.4. A multiplicity result for singular magnetic NLS equations 65

case A = 0. Furthermore, in [73] some examples of potentials satisfying(2.4.3) are given: when W is in the so-called Kato-Rellich class, namelyW ∈ Lq(RN ) + L∞(RN ), with q = 2 if N ≤ 3, q > 2 if N = 4 and q ≥ N/2if N ≥ 5, then the following property holds: for any ξ > 0 there existsαξ > 0 such that

∫W (x)|u|2 ≤ ξ||∇u||22 + αξ||u||

22 for any u ∈ H1(RN ) (for

the proof, see for instance [107]). This clearly implies (2.4.3). For example,the Coulomb potential |x|−1 is in the Kato-Rellich class, for N ≥ 3. Next,whenW ∈ LN/2(RN )∩Lβ(RN ) for some β > N/2, then the eigenvalue problem−∆u = λW (x)u, u ∈ D1,2(RN ), has the same properties as an eigenvalue prob-lem for −∆ in a bounded domain (see [49]). In particular, the first eigenvalueλ1(−∆,RN ,W ) is strictly positive and, as a consequence, (2.4.3) is satisfiedwith α1 = λ1(−∆,RN ,W ) and α2 = 0. Finally, when W (x) = |x|−2 (such apotential is none of the previous classes), Hardy inequality gives (2.4.3), withα1 = 4/(N − 2)2 and α2 = 0.We aim to obtain a multiplicity result by relating the number of solutions forthe problem

(~

i∇−A(x)

)2

u+ Vε(x)u = |u|p−2u in RN , (2.4.4)

with the topology of the set of global minima of V , in the semiclassical limitby using topological arguments (cf. [20], [22], [48]). Furthermore, we shallapply the result in [45] on the asymptotic behavior of the found solutions to(2.4.4).Before stating our main result, we need some notations. Let

V0 = infξ ∈RN

V (ξ), (2.4.5)

and

M = ξ ∈ RN : V (ξ) = V0. (2.4.6)

the set of global minima of V . For any δ > 0, let us denote Mδ = x ∈ RN :

d(x,M) ≤ δ.The key-point of our approach to (2.4.4) is that for any ξ ∈ M , the ellipticequation

(1

i∇−A(ξ)

)2

v + V0v = |v|p−2v, in RN (2.4.7)


is some kind of limit problem for (2.4.4). In fact, by the change of variabley = (x− ξ)/~ and for any ξ ∈ M , v~(y) = u~(~y + ξ) satisfies

− ∆yv~ −2

iA(~y + ξ)∇yv~(y) −

~

i(divxA)(~y + ξ)v~(y)

+ |A(~y + ξ)|2v~(y) + (V (~y + ξ)− ε(~)W (~y + ξ))v~(y) = |v~(y)|p−2v~(y).

Since A(~y + ξ) → A(ξ), V (~y + ξ) → V (ξ) = V0, W (~y + ξ) → W (ξ) andε(~) → 0 (direct consequence of (2.4.9)) as ~ → 0, v~ might tend to some v inweak-sense and v satisfies (2.4.7). Let v : R

N → C be a solution of (2.4.7) andset τξ(y) =

∑Nj=1Aj(ξ)yj , then A(ξ) = ∇τξ and v(y) = e−iτξ(y)v(y) satisfies

the complex equation

−∆v + V (ξ)v = |v|p−2v, in RN . (2.4.8)

As proved in [45], the magnetic potential A(x) only contributes to the phasefactor of the solutions of (2.4.4) as ~ small enough.

Remark 2.4.1. Let us recall that, if Y is a closed subset of a topological spaceX, catX(Y ) denotes Ljusternik-Schnirelman category of Y in X, i.e. the leastnumber of closed and contractible sets in X which cover Y (see [81], [98]).

Our main result is the following:

Theorem 2.4.2. Assume that A ∈ C1(RN ,RN ), V is a real continuousfunction in R

N satisfying

(H1) lim infx→+∞ V (x) > V0 > 0

and W is a real measurable function such that (2.4.3) holds.For any δ > 0 there exists ε∗∗(δ) > 0 such that, if

lim sup~→0

ε(~)

~2< ε∗∗(δ), (2.4.9)

then (2.4.4) has at least catMδ(M) solutions, for ~ sufficiently small.

2.5 Magnetic fields: the space HA

In this Section we recall some classical results on Schrodinger operators withmagnetic field, which are useful in the following.


We consider the spaceHA(RN ,C) consisting of all the functions u ∈ L2(RN ,C)with (∂j + iAj)u ∈ L2(RN ,C) for any j = 1, ......, N endowed with the norm

||u||2HA=

∫

RN

|(∇ + iA)u|2 dx+

∫

RN

|u|2 dx.

Remark 2.5.1. We remark that we do not assume that ∇u or Au are sep-arately in L2(RN ,C). Therefore, in general, there is no relationship betweenthe spaces HA(RN ,C) and H1(RN ,C), namely HA(RN ,C) is not a subset ofH1(RN ,C) or H1(RN ,C) is not a subset of HA(RN ,C) (see [55]).

Theorem 2.5.2. Let A : RN → R

N be in L2loc(R

N ) and let u ∈ H1A(RN ,C).

Then |u| ∈ H1(RN ,R) and the diamagnetic inequality

|∇|u|(x)| ≤ |(∇ + iA)u(x)|

holds for almost every x ∈ RN .

By the diamagnetic inequality, the following result follows (see [55]):

Theorem 2.5.3. The space C∞0 (RN ,C) is dense in H1

A(RN ,C).

2.6 The variational framework

Let E~ be the real Hilbert subspace defined by the closure of C∞0 (RN ,C) under

the scalar product

(u|v)~ ≡ Re

∫

RN

(D~uD~v + V (x)uv

)dx,

where D~u = (D~1u, ....., D

~

Nu) and D~j = i−1

~∂j − Aj(x). The norm inducedby the scalar product (·|·)~ is

||u||2~ ≡

∫

RN

(|D~u|2 + V (x)|u|2

)dx <∞.

Since A is real valued, it is easy to deduce that (see, for example, [66],[91]) forany u ∈ E~

~|∇|u|(x)| = ~

∣∣∣∣Re

(∇u

u

|u|

)∣∣∣∣ =∣∣∣∣Re

((~∇u− iAu)

u

|u|

)∣∣∣∣ ≤ |D~u(x)| (2.6.1)


a.e. in RN and |u| ∈ H1(RN ,R).

In E~ we define the functional

I~,ε(u) =

∫

RN

(|D~u|2 + Vε(x)|u|

2)dx.

Clearly, I~,ε(u) ≤ ||u||2~

for any u ∈ E~. Conversely, if (2.4.3) holds and0 < ~

2 ≤ V0α1α−12 , then for any u ∈ E~ we have

(1 − α1

ε(~)

~2

)||u||2~ ≤ I~,ε(u). (2.6.2)

Indeed,

∫

RN

W (x)|u|2 ≤ α1

∫

RN

|∇|u||2 +α2

V0

∫

RN

V (x)|u|2

≤α1

~2

[∫

RN

~2|∇|u||2 +

∫

RN

V (x)|u|2]

≤α1

~2

[∫

RN

(|D~u|2 + V (x)|u|2

)dx

]

=α1

~2||u||2~. (2.6.3)

As a consequence,

||u||2~ = I~,ε(u) + ε(~)

∫

RN

W (x)|u|2 dx ≤ I~,ε(u) + α1ε(~)

~2||u||2~

whence (2.6.2) follows. From (2.6.2), if

lim sup~→0

ε(~)~−2 < α−11 , (2.6.4)

there exist α0, ~∗0 > 0 such that

I~,ε(u) ≥ α0||u||2~ (2.6.5)

for any u ∈ E~, for any 0 < ~ < ~∗0. As a result, the space E~, endowed with

the norm ||u||2~,ε = I~,ε(u) equivalent to ||u||2

~, is a Hilbert space.

Furthermore, by (2.6.1) and (2.6.5) it results

||u||2~,ε ≥ α0||u||2~ ≥ min

~

2, V0

α0

(∫

RN

|∇|u||2 + |u|2 dx

). (2.6.6)


Let us consider the energy functional

J~,ε(u) =1

2||u||2~,ε −

1

p

∫

RN

|u|p dx, u ∈ E~.

For convenience we recall that for any u, v ∈ E~

⟨J ′

~,ε(u), v⟩

= Re

∫

RN

(D~uD~v + (V (x) − ε(~)W (x))uv − |u|p−2uv

)dx.

Furthermore, let us define the Nehari manifold of (2.4.4)

Σ~,ε = u ∈ E~ \ 0 : g~,ε(u) = 0 ,

where

g~,ε(u) ≡⟨J ′

~,ε(u), u⟩

= ||u||2~,ε −

∫

RN

|u|p dx.

Lemma 2.6.1. For any 0 < ~ < ~∗0 there exist σ~,ε, τ~,ε > 0 such that for any

u ∈ Σ~,ε

||u||~,ε ≥ σ~,ε,⟨g′~,ε(u), u

⟩≤ −τ~,ε. (2.6.7)

Proof. By (2.6.2), for any u ∈ Σ~,ε we have

0 < ||u||2~,ε =

∫

RN

|u|p dx ≤ C1

(∫

RN

|D~u|2 + V (x)|u|2 dx

)p/2

= C1||u||p~≤ C1

(1 − α1

ε(~)

~2

)−p/2

||u||p~,ε

and

0 < ||u||2~,ε

(1 − α1

ε(~)

~2

)p/2≤ C1||u||

p~,ε

(where C1 is a positive constant). This plainly implies the left-hand sideinequality in (2.6.7).Furthermore,

⟨g′~,ε(u), u

⟩= 2||u||2~,ε − p

∫

RN

|u|p dx

= (2 − p)||u||2~,ε ≤ (2 − p)σ~,ε

which gives the right-hand side inequality in (2.6.7).


By Lemma 2.6.1, Σ~,ε is a smooth manifold in E~. It is easy to see that J~,ε

is well defined and smooth on Σ~,ε; if u is a critical point of J~,ε on Σ~,ε, thenu is a weak solution for (2.4.4).

Remark 2.6.2. It is easy to see that for any v ∈ E~ \ 0 the ray Rt =tv : t ≥ 0 intersects the Nehari manifold Σ~,ε once and only once at θv (θ >0), where J~,ε(tv), t ≥ 0, achieves its maximum. Simple computations showthat critical points of f(t) ≡ J~,ε(tv) occur at, and only at, the intersectionsbetween the ray Rt and Σ~,ε (see [106], Lemma 2.1 for details).

Before beginning the next Sections, let us recall some known facts aboutthe equation with constant coefficients

−∆u+ λu = |u|p−2u in RN , (2.6.8)

where λ > 0. Let Σrλ be the Nehari manifold of (2.6.8), defined by λ,

Σrλ =

u ∈ H1(RN ,R) \ 0 :

∫

RN

(|∇u|2 + λ|u|2 − |u|p) dx = 0

and I the associated energy functional,

Iλ(u) =1

2

∫

RN

(|∇u|2 + λ|u|2) dx−1

p

∫

RN

|u|p dx, u ∈ Σrλ.

It is possible to prove that, for any λ > 0 Eq. (2.6.8) has a positive groundstate ω ≡ ω(λ), namely a solution satisfying

Iλ(ω) = infu∈Σr

λ

Iλ(u) ≡ mrλ (2.6.9)

(e.g. see [106], Proof of Lemma 2.2). By Gidas et al.[59], ω is spherically sym-metric about some point in R

N ; since (2.6.8) is invariant under translations,we can assume that ω is spherically symmetric about the origin. Moreover, ωis decreasing in r = |x| and it decays exponentially at infinity.

Remark 2.6.3. By Sobolev embedding Theorem the ground states in (2.6.9)are finite and positive. Moreover they depend continuously on the coefficientλ and satisfy a monotonicity property: if λ1 ≤ λ2 then mr

λ1≤ mr

λ2. We refer

to Lemmas 2.2 and 2.4 in [106] for some details.

2.7. The limit energy level 71

In a similar way we define the class

Σcλ =

u ∈ H1(RN ,C) \ 0 :

∫

RN

(|∇u|2 + λ|u|2 − |u|p) dx = 0

and denote by

mcλ ≡ inf

u∈Σcλ

Iλ(u). (2.6.10)

In [70], Lemma 7, it has been proved the following result.

Lemma 2.6.4. The equality mrλ = mc

λ =: mλ holds and the least energysolutions ωc ≡ ωc(λ) for mc

λ have the form eiσω(y − y0) for some y0 ∈ RN

and for some constant σ ∈ R.

In the case λ = V0, let ω (resp. ωc) be the least energy solution associatedto the energy functional IV0 on Σr

V0(resp. Σc

V0). In the following, we denote

I0 = IV0 , Σr0 = Σr

V0and Σc

0 = ΣcV0

. It is well known that ω is a positivesolution to the equation

−∆ω + V0ω = |ω|p−2ω in RN . (2.6.11)

We denote by mr0 (resp. mc

0) the least energy of I0(ω) (resp. I0(ωc)) on Σr0

(resp. Σc0). By Lemma 2.6.4, it results mr

0 = mc0 =: m0. By (H1), we can

choose V∞ ∈ R such that

lim inf|x|→+∞

V (x) ≥ V∞ > V0. (2.6.12)

Let us denote

m∞ := mrV∞ = mc

V∞ .

By Remark 2.6.3 and (2.6.12):

m0 < m∞. (2.6.13)

2.7 The limit energy level

In the present and the next Section, we shall introduce two maps Φ~ and βwhich permit to compare the topology of M to the topology of a suitablesublevel of the functional J~,ε, via Lemma 2.3.2.


Now let δ > 0 be fixed. Let η be a smooth nonincreasing cut-off function,defined in [0,+∞), such that η(t) = 1 if 0 ≤ t ≤ δ/2, η(t) = 0 if t ≥ δ,0 ≤ η ≤ 1 and |η′(t)| ≤ c for some c > 0. For any ξ ∈ M , let

ψ~,ξ(x) = η(|x− ξ|)ω

(x− ξ

~

)eiτξ(

x−ξ~

),

where ω is the positive ground state of (2.6.11), spherically symmetric aboutthe origin and τξ(y) =

∑Nj=1Aj(ξ)yj . For convenience we denote τ(y) = τξ(y).

By definition, ψ~,ξ has compact support for any ξ ∈ M , hence it belongs toE~. Let us define the map Φ~ : M → Σ~,ε by Φ~(ξ) = θ~ψ~,ξ, where θ~ > 0 issuch that θ~ψ~,ξ ∈ Σ~,ε (cf. Remark 2.6.2).

Lemma 2.7.1. Uniformly in ξ ∈ M , we have

lim sup~→0

~−NJ~,ε(Φ~(ξ)) ≤ m0. (2.7.1)

Proof. Let ξ ∈ M . First of all, let us prove that lim sup~→0 θ~ ≤ 1. By thedefinition of θ~ we have

θp~

∫

RN

|ψ~,ξ|p dx = θ2

~

∫

RN

(|D~ψ~,ξ|

2 + (V (x) − ε(~)W (x)) |ψ~,ξ|2)dx

≤ θ2~

∫

RN

(|D~ψ~,ξ|

2 + V (x)|ψ~,ξ|2)dx. (2.7.2)

By the change of variable y = (x − ξ)/~ and the exponential decay of w, wededuce

∫

RN

V (x)|ψ~,ξ|2 dx = ~

N

∫

RN

V (~x+ ξ)|ω(x)η(|~x|)|2 dx

= ~N

(∫

RN

V (ξ)|ω(x)|2 dx+ o(1)

)

= ~N

(∫

RN

V0|ω(x)|2 dx+ o(1)

)

as ~ → 0. Analogously,

∫

RN

|ψ~,ξ|p dx = ~

N

∫

RN

|ω(x)η(|~x|)|p dx

= ~N

(∫

RN

|ω(x)|p dx+ o(1)

)

2.7. The limit energy level 73

as ~ → 0. Furthermore, let M = max (||η||∞, ||∇η||∞). We have∫

RN

|D~ψ~,ξ|2 dx =

∫

RN

∣∣∣∣ i−1

~∇η(|x− ξ|)ω

(x− ξ

~

)eiτ((x−ξ)/~)

+ i−1η(|x− ξ|)∇ω

(x− ξ

~

)eiτ((x−ξ)/~)

+ η(|x− ξ|)(A(ξ) −A(x))ω

(x− ξ

~

)eiτ((x−ξ)/~)

∣∣∣∣2

dx

= ~N

∫

RN

∣∣ i−1~∇η(|~x|)ω(x) + i−1η(|~x|)∇ω(x)

+ η(|~x|)(A(ξ) −A(~x+ ξ))ω(x)|2 dx.

By the above identity we derive that∫

RN

|D~ψ~,ξ|2 dx = ~

N

∫

RN

|∇ω|2 dx+ I,

where

I ≤ 2~N+1M2

∫

δ2≤ |~y| ≤ δ

|ω(y)||∇ω(y)| dy + ~N+2M2

×

∫

δ2≤ |~y| ≤ δ

|ω(y)|2 dy + 2~NM2

∫

|~y| ≤ δ|ω(y)|2|A(ξ) −A(ξ + ~y)| dy

+ 2~NM2

∫

|~y| ≤ δ|ω(y)||∇ω(y)||A(ξ) −A(ξ + ~y)| dy

+ 2~N+2M2

∫

|~y| ≤ δ|ω(y)|2|A(ξ) −A(ξ + ~y)| dy.

Using the exponential decay of ω and |∇ω|, we can infer∫

|~y| ≤ δ|ω(y)|2|A(ξ) −A(ξ + ~y)| dy ≤ C~

∫

RN

|ω(y)|2|y| dy = O(~).

Using the exponential decay of ω and |∇ω|, we control the other terms in asimilar way and we conclude

∫

RN

|D~ψ~,ξ|2 dx = ~

N

(∫

RN

|∇ω|2 dx+ o(1)

).

Hence by (2.7.2) we obtain that

θp−2~

~N

(∫

RN

|ω(x)|p dx+ o(1)

)≤ ~

N

(∫

RN

(|∇ω|2 + V0|ω(x)|2) dx+ o(1)

).


Since p > 2 and∫

RN (|∇ω|2 +V0|ω(x)|2) dx =∫

RN |ω(x)|p dx, we conclude thatlim sup~→0 θ~ ≤ 1. Furthermore by a change of variable y = (x − ξ)/~ wehave

~−NJ~,ε(Φ~(ξ)) =

(1

2−

1

p

)θp

~

∫

RN

|ω(x)η(|~x|)|p dx

≤ I0(ω) + o(1) = m0 + o(1).

Thus (2.7.1) is proved moreover, the limit is uniform in ξ, being M a compactset.

2.8 The barycentre map

Let ρ > 0 be such that Mδ ⊂ Bρ =x ∈ R

N : |x| ≤ ρ. Let χ : R

N → RN

be defined by χ(x) = x for |x| ≤ ρ and χ(x) = ρx/|x| for |x| ≥ ρ. Finally,let us define β : Σ~,ε → R

N by

β(u) =

∫RN χ(x)|u(x)|2 dx∫

RN |u(x)|2 dx, u ∈ Σ~,ε.

Arguing as in the proof of Lemma 2.7.1, it is easy to see that

β(Φ~(ξ)) =

∫RN χ(x)|Φ~(ξ)(x)|

2 dx∫RN |Φ~(ξ)(x)|2 dx

=

∫RN χ(~x+ ξ)|ω(x)η(|~x|)|2 dx∫

RN |ω(x)η(|~x|)|2 dx

= ξ +

∫RN (χ(~x+ ξ) − ξ)|ω(x)η(|~x|)|2 dx∫

RN |ω(x)η(|~x|)|2 dx= ξ + o(1), (2.8.1)

as ~ → 0, uniformly for ξ ∈ M .

Lemma 2.8.1. For any δ > 0 there exists ε∗1(δ) > 0 such that, if

lim sup~→0

ε(~)

~2< ε∗1(δ),

then there exist ~∗1, k

∗1 > 0 such that 0 < ~ < ~

∗1, u ∈ Σ~,ε and J~,ε(u) ≤

~N (m0 + k∗1) imply β(u) ∈Mδ.

2.8. The barycentre map 75

Proof. By contradiction, let us assume that for some δ > 0, we can findεm ≥ 0 such that εm → 0 as m→ ∞, lim sup~→0 ε(~)~−2 = εm and the claimof Lemma 2.8.1 does not hold.For ~ small we have ε(~)~−2 < εm + 1

m and, by (2.6.2),

(1 − α1

(εm +

1

m

))||u||2~ ≤ ||u||2~,ε. (2.8.2)

Let ~n, kn → 0+ as n → ∞ and un ∈ Σ~n,εn be such that J~n,εn(un) ≤~Nn (m0 + kn) and β(un) /∈Mδ.

In order to deduce a contradiction, it suffices to find points ξn ∈ Mδ suchthat

limn→∞

|β(un) − ξn| = 0, (2.8.3)

possibly up to a subsequence. For any n, let vn(x) = un(~nx); we shall provethe following result:

Claim 2.8.2. There exists ξn ⊂ RN such that ~nξn converges to a point

ξ ∈ M as n → +∞. Furthermore, up to a subsequence, |vn(y + ξn)| con-verges strongly in H1(RN ,R) to ω, namely the positive ground state of (2.6.11),radially symmetric around the origin.

Proof of the Claim 2.8.2. The proof of Claim 2.8.2 is based on Concentration-Compactness Lemma by Lions (cf. [76, 77]).For any n ∈ N let us consider the measure

µn(Ω) =p− 2

2p

∫

Ω

(|∇|vn||

2 + V (~nx)|vn|2)dx.

Since un ∈ Σ~n,εn , by (2.8.2), we have

p− 2

2p

(1 − α1

(εm +

1

m

))||un||

2~n

≤p− 2

2p||un||

2~n,εn

≤ ~Nn (m0 + kn),

(2.8.4)where

||un||2~n,εn

= ~Nn

∫

RN

(|(∇− iA(~nx))vn|

2 + (V (~nx) − ε(~n)W (~nx))|vn|2)dx.

(2.8.5)


By (2.8.4) and (2.8.5) we obtain

0 ≤ µn(RN ) =

p− 2

2p

∫

RN

(|∇|vn||

2 + V (~nx)|vn|2)dx

≤m0 + kn(

1 − α1

(εm + 1

m

)) .

Therefore, as n,m→ ∞, µn converges to some m ≤ m0, up to a subsequence.Furthermore, m > 0 since

µn(RN ) =

p− 2

2p~−Nn

∫

RN

(~

2n|∇|un||

2 + V (x)|un|2)dx > 0.

Now by the Concentration-Compactness Lemma by Lions there exists a subse-quence of µn (denoted in the same way) satisfying one of the three followingpossibilities:

Vanishing: For all ρ > 0

limn→∞

supξ∈RN

∫

Bρ(ξ)dµn = 0.

Dichotomy: There exist a constant m ∈ (0, m), sequences ξn, ρn suchthat |ξn|, ρn → +∞ and two nonnegative measures µ1

n and µ2n satisfying the

following:

0 ≤ µ1n+µ2

n ≤ µn, µ1n(R

N ) → m, µ2n(R

N ) → m−m supp(µ1n) ⊂ Bρn(ξn),

supp(µ2n) ⊂ R

N \B2ρn(ξn).

Compactness: There exists a sequence ξn with the following property: forany ε > 0 there exists ρ > 0 such that

∫

Bρ(ξn)dµn ≥ m− ε. (2.8.6)

It is easy to see that vanishing cannot occur. Otherwise by Lions [77] (see alsoLemma 2.18 in [50]) we have vn → 0 as n → +∞ in Lr for each 2 < r < 2∗.Then ∫

RN

|vn|p dx→ 0


as n→ +∞. Since by (2.8.4) and (2.8.5)

µn(RN ) ≤

p−22p

∫RN |vn|

p dx(1 − α1

(εm + 1

m

)) ,

we infer µn(RN ) → 0 as n,m→ +∞ and this contradicts µn(R

N ) → m > 0.We show now that dichotomy does not occur. Similar arguments are used in[106] in the case A = 0. Otherwise take χn ∈ C1

0 (RN ) such that χn ≡ 1 onBρn(ξn) and χn ≡ 0 on Bc

2ρn(ξn) and 0 ≤ χn ≤ 1, |∇χn| ≤ 2/ρn. Let us

denote |vn| = φn and set

φn = χnφn + (1 − χn)φn ≡ φ1n + φ2

n,

where φ1n and φ2

n are defined in the last equality. Then

p− 2

2p

∫

RN

(|∇φ1

n|2 + V (~nx)|φ

1n|

2)dx

≥ µn(Bρn(ξn)) ≥ µ1n(Bρn(ξn)) = µ1

n(RN ) → m (2.8.7)

and

p− 2

2p

∫

RN

(|∇φ2n|

2 + V (~nx)|φ2n|

2) dx

≥ µn(Bc2ρn

(ξn)) ≥ µ2n(B

c2ρn

(ξn)) = µ2n(R

N ) → m−m. (2.8.8)

Now let Ωn = B2ρn(ξn) \Bρn(ξn), we derive

p− 2

2p

∫

Ωn

(|∇φ2

n|2 + V (~nx)|φ

2n|

2)dx

= µn(Ωn) = µn(RN ) − µn(Bρn(ξn)) − µn(B

c2ρn

(ξn))

≤ µn(RN ) − µ1

n(Bρn(ξn)) − µ2n(B

c2ρn

(ξn)) → 0. (2.8.9)

Now we observe that∫

RN

(|∇φn|

2 + V (~nx)|φn|2)dx

=

∫

RN

(|∇φ1n|

2 + V (~nx)|φ1n|

2) dx

+

∫

RN

(|∇φ2n|

2 + V (~nx)|φ2n|

2) dx+An, (2.8.10)


where An → 0 as n→ +∞ by (2.8.9).Moreover, by (2.8.9) and Sobolev imbedding we derive

∫Ωn

|φn|p dx → 0 as

n→ +∞. Consequently, we get

∫

RN

|φn|p dx =

∫

RN

|φ1n + φ2

n|p dx

=

∫

Bρn (ξn)|φ1n|p dx+

∫

Bc2ρn

(ξn)|φ2n|p dx+

∫

Ωn

|φn|p dx

=

∫

RN

|φ1n|p dx+

∫

RN

|φ2n|p dx+ o(1). (2.8.11)

Now by (2.8.7)-(2.8.8) and (2.8.9)-(2.8.10) we have

2p

p− 2m = lim

n→+∞

∫

RN

(|∇φn|

2 + V (~nx)|φn|2)dx

= limn→+∞

(∫

RN

(|∇φ1n|

2 + V (~nx)|φ1n|

2) dx

+

∫

RN

(|∇φ2n|

2 + V (~nx)|φ2n|

2) dx+ o(1)

)

≥ limn→+∞

∫

RN

(|∇φ1n|

2 + V (~nx)|φ1n|

2)) dx

+ limn→+∞

∫

RN

(|∇φ2n|

2 + V (~nx)|φ2n|

2)) dx

≥2p

p− 2(m+ (m−m)) =

2p

p− 2m.

Therefore,p− 2

2p

∫

RN

(|∇φ1n|

2 + V (~nx)|φ1n|

2) dx→ m

andp− 2

2p

∫

RN

(|∇φ2n|

2 + V (~nx)|φ2n|

2) dx→ m−m.

Now let us denote

A1n =

∫

RN

(|∇φ1n|

2 + V (~nx)|φ1n|

2) dx−

∫

RN

|φ1n|p dx

and

A2n =

∫

RN

(|∇φ2n|

2 + V (~nx)|φ2n|

2) dx−

∫

RN

|φ2n|p dx.


Since

∫

RN

(|∇φn|

2 + V (~nx)|φn|2)dx ≤

∫RN |φn|

p dx(1 − α1

(εm + 1

m

)) ,

by (2.8.10) and (2.8.11) as m→ ∞, we deduce that

A1n +A2

n ≤ o(1). (2.8.12)

We shall prove that (2.8.12) is not true.After passing to a subsequence we assume that A1

n ≤ 0. Fix θ1n such that

θ1nφ

1n ∈ Σr

0. Then

θ1np−2

∫

RN

|φ1n|p dx =

∫

RN

(|∇φ1n|

2 + V0|φ1n|

2) dx

≤

∫

RN

(|∇φ1n|

2 + V (~nx)|φ1n|

2) dx

≤

∫

RN

|φ1n|p dx.

Therefore θ1n ≤ 1 and hence

m0 ≤p− 2

2p(θ1n)

2

∫

RN

(|∇φ1n|

2 + V0|φ1n|

2) dx

≤p− 2

2p

∫

RN

(|∇φ1n|

2 + V (~nx)|φ1n|

2) dx→ m < m ≤ m0.

This is absurd. Arguing as before we can conclude that also A2n ≤ 0, passing

to a subsequence.Now we assume that passing to a subsequence A1

n > 0 and A2n > 0. Therefore

by (2.8.12), we infer A1n = o(1) and A2

n = o(1).If θ1

n ≤ 1 + o(1) we can argue as before and we get a contradiction. Now weassume that θ1

n → θ0 > 1 as n→ +∞.Then we have

∫RN |φ1

n|p dx tends to a positive limit as n → +∞. Otherwise

we have ∫

RN

(|∇φ1n|

2 + V (~nx)|φ1n|

2) dx→ 0

as n→ +∞ and then m = 0. This is absurd. Now observe that

0 = limn→+∞

A1n ≥ lim

n→+∞((θ1

n)p−2 − 1)

∫

RN

|φ1n|p dx

= ((θ0)p−2 − 1) lim

n→∞

∫

RN

|φ1n|p dx > 0.


We find a contradiction.Then compactness occurs or, in other words, the sequence µn is tight. Letus prove that the sequence ~nξn is bounded, assuming for the moment V∞ <+∞. Actually, this is the most interesting case, and we shall indicate belowhow to modify the argument if V∞ = +∞.We claim now that ~nξn is a bounded sequence. Otherwise there exists asubsequence still denoted by ~nξn such that |~nξn| → +∞ as n→ +∞.Now let us denote wn(x) = vn(x + ξn). By (2.8.4) |wn| is bounded inH1(RN ,R) and satisfies

∫

RN

|wn|p ≥

(1 − α1

(εm +

1

m

))∫

RN

(|∇|wn||

2 + V (~nx+ ~nξn)|wn|2).

(2.8.13)Up to a subsequence, we may assume that |wn| weakly converges to a nonneg-ative v ∈ H1(RN ,R). Furthermore as µn(R

N ) → m, (2.8.6) implies

∫

RN\Bρ(0)

(|∇|wn||

2 + V0|wn|2)dx ≤ ε

for n large, thus |wn| converges strongly in Lr(RN ,R) to v with 1 ≤ r < 2∗.By again (2.8.6) we have v 6= 0.Now let us denote θ∞ > 0 such that θ∞v ∈ Σr

V∞, the Nehari manifold defined

by V∞. So that by (2.8.13) we deduce

∫

RN

vp ≥ lim supn→+∞

∫

RN

|wn|p

≥ lim supn,m→+∞

[(1 − α1

(εm +

1

m

))∫

RN

(|∇|wn||

2 + V (~nx+ ~nξn)|wn|2)]

≥ lim infn,m→+∞

[(1 − α1

(εm +

1

m

))∫

RN

(|∇|wn||

2 + V (~nx+ ~nξn)|wn|2)]

≥ lim infn→+∞

∫

RN

|∇|wn||2 dx+ lim inf

n→+∞

∫

RN

V (~nx+ ~nξn)|wn|2

≥

∫

RN

(|∇v|2 + V∞v

2)

= θp−2∞

∫

RN

vp

so that θ∞ ≤ 1. Therefore by (2.8.4) we have


m∞ = mV∞ ≤ θ2∞p− 2

2p

∫

RN

(|∇v|2 + V∞|v|2

)dx

≤p− 2

2plim infn→+∞

∫

RN

(|∇|wn||

2 + V (~nx+ ~nξn)|wn|2)dx

≤ lim infn,m→+∞

(m0 + kn(

1 − α1

(εm + 1

m

)))

= lim infn→+∞

(m0 + kn).

As n,m→ +∞, the before inequality contradicts condition (2.6.13). Thereforewe can assume that ~nξn → ξ.If V∞ = m∞ = +∞, we can choose V > 0 such that mr

V> m0 and repeat the

argument above with V∞ replaced by V . In conclusion the sequence ~nξn isbounded and it converges to some ξ (up to subsequence).Now we aim to prove that ξ ∈M .Firstly, let θ > 0 such that θv ∈ Σr

0, where Σr0 is the Nehari manifold associated

to (2.6.11) (cf. Section 2.6). Then by (2.8.13) we have

∫

RN

vp ≥ lim supn→+∞

∫

RN

|wn|p

≥ lim supn,m→+∞

[(1 − α1

(εm +

1

m

))∫

RN

(|∇|wn||

2 + V (~nx+ ~nξn)|wn|2)]

≥ lim infn,m→+∞

[(1 − α1

(εm +

1

m

))∫

RN

(|∇|wn||

2 + V (~nx+ ~nξn)|wn|2)]

≥ lim infn→+∞

∫

RN

|∇|wn||2 dx+ lim inf

n→+∞

∫

RN

V (~nx+ ~nξn)|wn|2

≥

∫

RN

(|∇v|2 + V∞v

2)

= θp−2

∫

RN

vp

so that θ ≤ 1. Hence, as in the previous calculations, we conclude


m0 ≤ mV (ξ) ≤ θ2 p− 2

2p

∫

RN

(|∇v|2 + V (ξ)v2

)dx

≤p− 2

2p

∫

RN

(|∇v|2 + V (ξ)v2

)dx

≤p− 2

2plim infn→+∞

∫

RN

(|∇|wn||

2 + V (~nx+ ~nξn)|wn|2)dx

≤ lim infn,m→+∞

(m0 + kn(

1 − α1

(εm + 1

m

)))

= lim infn→+∞

(m0 + kn). (2.8.14)

Therefore we deduce that θ = 1 and m0 = mV (ξ). Hence, we have ξ ∈M andv is a positive ground state of (2.6.11) which is radially symmetric about theorigin by (2.8.6). Thus v = ω. The strong convergence |wk| → v in Lr(RN ,R)(1 ≤ r ≤ 2∗) and (2.8.14) give

lim infn→+∞

∫

RN

(|∇|wn||

2 + V (~nx+ ~nξn)|wn|2)

=

∫

RN

(|∇v|2 + V (ξ)v2

)

=

∫

RN

(|∇v|2 + V0v

2).

This implies that |wn| → v in H1(RN ,R) as n→ +∞.

By Claim 2.8.2 we have ~nξn → ξ ∈ M and thus we can assume ξn ≡ ~nξn ∈Mδ, at least for n large. Moreover,

β(un) =

∫RN χ(x)|un(x)|

2 dx∫RN |un(x)|2 dx

=

∫RN χ(~nx)|vn(x)|

2 dx∫RN |vnx)|2 dx

=

∫RN χ(~nx+ ξn)|vn(x+ ξn)|

2 dx∫RN |vn(x+ ξn)|2 dx

= ξn +

∫RN (χ(~nx+ ξn) − ξn) |vn(x+ ξn)|

2 dx∫RN |vn(x+ ξn)|2 dx

. (2.8.15)

Now, by Claim 2.8.2, we have |vn(·+ξn)| → ω strongly in H1(RN ,R), and thusapplying Lebesgue Theorem to (2.8.15) we deduce (2.8.3) then a contradiction.

2.9. The Palais-Smale condition 83

2.9 The Palais-Smale condition

In order to apply the global variational arguments, we need J~,ε to satisfyPalais-Smale condition on Σ~,ε. Nevertheless, we shall prove that Palais-Smalecondition holds in a suitable sublevel, related to the ground state energy ”atinfinity”. We recall that here we do not require any assumptions (K1) and(K2) in [70].

Lemma 2.9.1. Assume

lim sup~→0

ε(~)

~2<

1

α1

(1 −

m0

m∞

). (2.9.1)

Then there exist ~∗2 > 0 and k∗2 ∈ (0,m∞ −m0) such that the functional J~,ε

satisfies Palais-Smale condition on the sublevel

u ∈ Σ~,ε : J~,ε(u) < ~

N (m0 + k∗2),

for any 0 < ~ < ~∗2.

Proof. For convenience, throughout the proof we shall denote by νn a sequence(not necessarily the same) that goes to zero as n→ ∞; moreover, C will denotea positive constant whose value changes as the case may be.To begin with, let ε0 the left-hand side of (2.9.1), let C ∈ (m0, (1−α1ε0)m∞)and fix η0 > 0 such that

C < (1 − α1(ε0 + η0))p

p−2m∞; (2.9.2)

let ~∗2 ∈ (0, ~∗

0) be such that ε(~) < (ε0 + η0)~2 for any 0 < ~ < ~

∗2 and assume

that for any such ~ Lemma 2.7.1 applies. Furthermore, the sublevel definedabove is not empty, since

inf J~,ε(u) : u ∈ Σ~,ε < ~Nm∞. (2.9.3)

Indeed, if (2.9.3) were not true, then m∞ ≤ ~−Nn inf J~n,εn(u) : u ∈ Σ~n,εn

along some sequence ~n → 0. Lemma 2.7.1 therefore implies m∞ ≤ m0 + νn,which contradicts condition (2.6.13).Next, let

λ < ~N C (2.9.4)

and let un be a Palais-Smale sequence for J~,ε on Σ~,ε at level λ, namely

J~,ε(un) = λ+ νn, J ′~,ε|Σ~,ε(un) = νn in E−1

~. (2.9.5)


As un ∈ Σ~,ε and by (2.9.5) we deduce un is bounded in E~ and, up toa subsequence, it has a weak limit u ∈ E~; we have to prove that u ∈ Σ~,ε

and that un → u strongly in E~. First of all remark that the left-hand sideequality in (2.9.5) implies J ′

~,ε(un) = νn, namely un is a Palais-Smale sequencefor the uncostrained functional. Indeed, J ′

~,ε(un) = νn+µng′~,ε(un) for suitable

µn; the boundedness of un implies 0 = g~,ε(un) =⟨J ′

~,ε(un), un

⟩= νn +

µn

⟨g′

~,ε(un), un

⟩; then (2.6.7) gives νn = |µn

⟨g′

~,ε(un), un

⟩| ≥ µnτ~,ε, thus

µn = νn and J ′~,ε(un) = νn, g

′~,ε(un) being bounded. As a consequence

νn =⟨J ′

~,ε(un), u⟩

= Re

∫

RN

(D~un ·D~u+ Vε(x)unu

)dx− Re

∫

RN

|un|p−2unu dx

= νn + ||u||2~,ε −

∫

RN

|u|p dx,

whence u ∈ Σ~,ε. In order to prove strong convergence it suffices to show thatfor any δ > 0 there exists R > 0 such that

∫

|x|≥R

(|D~un|

2 + V (x)|un|2)dx < δ for any n ≥ R. (2.9.6)

Indeed, (2.9.6) implies

∫

|x|≥R

(~

2|∇|un||2 + V (x)|un|

2)dx < δ for any n ≥ R.

Note that by (2.6.1) we get that un is also bounded in H1(K) for any boundeddomain K, and hence converges strongly in Lr(K), 2 ≤ r < 2∗.For any R > 0, by (2.6.1) and (2.6.6) we have


∣∣∣∣∣

∫

|x|≤R(|un|

p−2un − |u|p−2u)v dx

∣∣∣∣∣

p

≤

[∫

|x|≤R(||un|

p−2un − |u|p−2u|)p

p−1 dx

]p−1(∫

|x|≤R|v|p dx

)

≤ C

[∫

|x|≤R(||un|

p−2un − |u|p−2u|)p

p−1 dx

]p−1

×

(∫

|x|≤R(~2|∇|v||2 + V (x)|v|2) dx

)p/2

≤ C

[∫

|x|≤R(||un|

p−2un − |u|p−2u|)p

p−1 dx

]p−1

×

(∫

RN

(|D~v|2 + V (x)|v|2) dx

)p/2

≤ C

[∫

|x|≤R(||un|

p−2un − |u|p−2u|)p

p−1 dx

]p−1

α−p/20 ||v||p

~,ε

≤ o(1)||v||p~,ε (2.9.7)

and, if (2.9.6) holds,

(∫

|x|>R|un|

p−2unv dx

)p

≤

(∫

|x|>R|un|

p dx

)p−1(∫

|x|>R|v|p dx

)

≤ C

(∫

|x|>R(~2|∇|un||

2 + V (x)|un|2) dx

) p(p−1)2

×

(∫

|x|>R(~2|∇|v||2 + V (x)|v|2) dx

)p/2

≤ C

(∫

|x|>R(~2|∇|un||

2 + V (x)|un|2) dx

) p(p−1)2

α−p/20 ||v||p

~,ε

≤ Cδp(p−1)

2 ||v||p~,ε. (2.9.8)


Moreover, (2.9.6) implies

∫

|x|≥R

(~

2|∇|u||2 + V (x)|u|2)dx ≤ δ

which in turns gives for∫|x|>R |u|p−2uv dx the same estimate as in (2.9.8). By

(2.9.7) and (2.9.8) and analogous relations for the other nonlinear term, weobtain

∣∣∣∣∫

RN

(D~un ·D~v + (V (x) − ε(~)W (x))unv − |u|p−2uv

)dx

∣∣∣∣

≤

∣∣∣∣∫

RN

(D~un ·D~v + Vε(x)unv − |un|

p−2unv)dx

∣∣∣∣

+

∣∣∣∣∫

RN

(|un|


∣∣∣∣

≤ |⟨J ′

~,ε(un), v⟩| +

∣∣∣∣∣

∫

|x|≤R

(|un|


∣∣∣∣∣

+

∫

|x|>R

(|un|

p−1 + |u|p−1)|v| dx

≤ νn||v||~,ε + C1δp−12 ||v||~,ε. (2.9.9)

Now taking into account that u ∈ Σ~,ε and by (2.9.9) we deduce

||un||2~,ε =

∫

RN

(|D~un|2 + Vε(x)|un|

2) dx =

∫

RN

|u|p dx+ νn +O(δp−12 )

= ||u||2~,ε + νn +O(δp−12 ).

As δ can be chosen arbitrarily small (and the constants in (2.9.8) and (2.9.9)are independent of R ), we conclude ||un||

2~,ε → ||u||2

~,ε as n → ∞, which im-plies strong convergence.We still have to prove (2.9.6). By contradiction assume that for some subse-quence unk

and some δ0 > 0

∫

|x|≥k

(|D~unk

|2 + V (x)|unk|2)dx ≥ δ0 (2.9.10)

for any k. By the choice of λ and Remark 2.6.3 we can choose η > 0 such that~−Nλ < m(V∞−η) ≡ mη. Let R(η) > 0 be such that


V (x) ≥ V∞ − η, for |x| ≥ R(η)

(with no restrictions, assume R(η) is an integer). For any r > 0, let usconsider the annulus Ar =

x ∈ R

N : r ≤ |x| ≤ r + 1. We claim that there

exists r > R(η) such that

∫

Ar

(|D~unk

|2 + V (x)|unk|2)dx < η (2.9.11)

for infinitely many k. If this is not the case, for any integer m ≥ R(η) thereexists an integer ν(m) such that

∫

Am

(|D~unk

|2 + V (x)|unk|2)dx ≥ η

for any k ≥ ν(m). Plainly, we can assume that the sequence ν(m) is non-decreasing. Therefore, by (2.6.6), for any integer m ≥ R(η) there exists aninteger ν(m) such that

||unk||2~,ε =

∫

RN

(|D~unk

|2 + Vε(x)|unk|2)dx

≥ α0

∫

R(η)≤|x|≤m

(|D~unk

|2 + V (x)|unk|2)dx ≥ α0(m−R(η))η

for any k ≥ ν(m), a contradiction since ||unk||~,ε ≤ C.

Now, let us fix r = r(η) > R(η) so that (2.9.11) holds; then, up to a subse-quence, we have

∫

Ar

(|D~unk

|2 + V (x)|unk|2)dx < η (2.9.12)

for any k. Let ρ ∈ C∞(RN , [0, 1]) satisfy ρ(x) = 1 for |x| ≤ r, ρ(x) = 0 for|x| ≥ r+1 and |∇ρ(x)| ≤ 2 for any x ∈ R

N . Let vk = ρunkand wk = (1−ρ)unk

,for any k. It is not difficult to see that

∣∣⟨J ′~,ε(unk

), vk⟩−⟨J ′

~,ε(vk), vk⟩∣∣ ≤ C1

∫

Ar

(|D~unk

|2 + V (x)|unk|2)dx

(2.9.13)

and


∣∣⟨J ′~,ε(unk

), wk⟩−⟨J ′

~,ε(wk), wk⟩∣∣ ≤ C2

∫

Ar

(|D~unk

|2 + V (x)|unk|2)dx

(2.9.14)where C1 and C2 are positive constants which do not depend on r. Firstly, weprove (2.9.13).∣∣⟨J ′

~,ε(unk, vk⟩−⟨J ′

~,ε(vk), vk⟩∣∣

=

∣∣∣∣Re

∫

RN

(D~unk

·D~vk + Vε(x)unkvk − |unk

|p−2unkvk

)dx

−

∫

RN

(|D~vk|

2 + Vε(x)|vk|2 − |vk|

p)dx

∣∣∣∣

=

∣∣∣∣Re

∫

Ar

(D~unk

·D~(ρunk) + Vε(x)unk

(ρunk) − |unk

|p−2unk(ρunk

))dx

−

∫

Ar

(|D~(ρunk

)|2 + Vε(x)ρ2|unk

|2 − ρp|unk|p)dx

∣∣∣∣

=

∣∣∣∣Re

∫

Ar

D~unk·D~(ρunk

) dx+

∫

Ar

(Vε(x)|unk

|2 − |unk|p)ρ dx

−

∫

Ar

|D~(ρunk)|2 dx−

∫

Ar

(Vε(x)|unk

|2 − ρp−2|unk|p)ρ2 dx

∣∣∣∣ . (2.9.15)

Taking into account that |∇ρ| ≤ 2 on RN , it is easy to derive that

∫

Ar

|D~(ρunk)|2 dx =

∫

Ar

N∑

j=1

∣∣∣∣D~

j unkρ+

~

i

∂ρ

∂xjunk

∣∣∣∣2

dx

≤ C

∫

Ar

(|D~unk

|2 + V (x)|unk|2)dx, (2.9.16)

where C is a positive constant which do not depend on r. Analogously wehave

∣∣∣∣Re

∫

Ar

D~unkD~(ρunk

)

∣∣∣∣ =

∣∣∣∣∣∣Re

∫

Ar

N∑

j=1

(D~

j unk

(D~j unk

ρ+~

i

∂ρ

∂xjunk

))∣∣∣∣∣∣

=

∫

Ar

N∑

j=1

∣∣∣∣|D~

j unk|2ρ+ ~iD~

j unk

∂ρ

∂xjunk

∣∣∣∣

≤ C ′

∫

Ar

(|D~unk

|2 + V (x)|unk|2), (2.9.17)


where C ′ is a positive constant which do not depend on r.By (2.9.15)-(2.9.17) we easily deduce (2.9.13). Inequality (2.9.14) follows bysimilar arguments. Thus, by (2.9.5) and (2.9.12) we deduce

⟨J ′

~,ε(vk), vk⟩

= O(η) + νk,⟨J ′

~,ε(wk), wk⟩

= O(η) + νk. (2.9.18)

This implies

J~,ε(vk) ≥

(1

2−

1

p

)||vk||

2~,ε +O(η) + νk,

thus (2.9.5) and (2.9.18) yield

λ+νk = J~,ε(unk) = J~,ε(vk)+J~,ε(wk)+O(η) ≥ J~,ε(wk)+O(η)+νk. (2.9.19)

By (2.9.10) and the definition of wk:

∫

RN

(|D~wk|

2 + V (x)|wk|2)dx ≥ δ0 +O(η). (2.9.20)

Let θk > 0 be such that θkwk ∈ Σ~,ε; (2.9.20) implies θk bounded, so that(2.9.18) gives

θk = 1 +O(η) + νk (2.9.21)

(up to a subsequence); as a consequence

J~,ε(θkwk) = J~,ε(wk) +O(η) + νk. (2.9.22)

Let wk(x) = θkwk(~x) and let θk such that θk|wk| ∈ ΣrV∞−η, the Nehari

manifold defined by V∞ − η. From (2.6.1) and (2.6.2)


θp−2k ~

N

∫

RN

|wk|p dx = ~

N

∫

RN

(|∇|wk||

2 + (V∞ − η)|wk|2)dx

=

∫

RN

(~

2|∇(θk|wk|)|2 + (V∞ − η)|θkwk|

2)dx

≤

∫

RN

(|D~(θkwk)|

2 + V (x)|θkwk|2)dx

≤1

(1 − α1(ε0 + η0))

∫

RN

(|D~(θkwk)|

2 + Vε(x)|θkwk|2)dx

=1

(1 − α1(ε0 + η0))

∫

RN

|θkwk|p dx

=1

(1 − α1(ε0 + η0))~N

∫

RN

|wk|p dx (2.9.23)

it follows θk ≤ 1

(1−α1(ε0+η0))1

p−2. Now let us denote mη ≡ mr

V∞−η. Thus by

(2.9.19) and (2.9.22) again

~Nmη ≤ ~

N θ2k

p− 2

2p

∫

RN

(|∇|wk||

2 + (V∞ − η)w2k

)dx

≤ θ2k

p− 2

2p

∫

RN

(~

2|∇(θk|wk|)|2 + V (x)|θkwk|

2)dx

≤1

(1 − α1(ε0 + η0))2

p−2

p− 2

2p

∫

RN

(|D~(θkwk)|

2 + V (x)|θkwk|2)dx

=1

(1 − α1(ε0 + η0))p

p−2

p− 2

2p

∫

RN

(|D~(θkwk)|

2 + Vε(x)|θkwk|2)dx

=1

(1 − α1(ε0 + η0))p

p−2

J~,ε(θkwk)

=1

(1 − α1(ε0 + η0))p

p−2

(J~,ε(wk) +O(η) + ǫk)

≤1

(1 − α1(ε0 + η0))p

p−2

(λ+O(η) + νk). (2.9.24)

Letting k → ∞ and η → 0, (2.9.2) yields ~N C < λ which contradicts (2.9.4).

2.10. Proof of the main result 91

2.10 Proof of the main result

In order to compare the topology of M and the topology of a suitable energysublevel we will use the maps Φ~ and β introduced in Sections 2.7 and 2.8. Fur-thermore, in order to prove Theorem 2.4.2, we shall apply Theorem 2.3.1, anabstract multiplicity result for critical points involving Ljusternik-Schnirelmancategory recalled in Section 2.3.

Proof of Theorem 2.4.2. Let δ > 0 be fixed and ε∗1(δ) be as in Lemma2.8.1. Let

ε∗∗(δ) = min

1

α1

(1 −

m0

m∞

), ε∗1(δ)

and assume that (2.4.9) holds. Let 0 < ~∗ ≤ min ~∗

i : i = 1, 2 and k∗ =min k∗i : i = 1, 2, the constants ~

∗i , k

∗i being defined in Lemmas 2.8.1 and

2.9.1. Let 0 < ~ < ~∗; we can assume that a(~) ≡ ~

N (m0 +k∗) is not a criticalvalue for J~,ε on Σ~,ε. For convenience, we set Σ~,ε = u ∈ Σ~,ε : J~,ε(u) ≤ a(~).If ~

∗ is small enough, Lemma 2.9.1 guarantees that Palais-Smale conditionholds in a sublevel containing Σ~,ε and Lemma 2.7.1 gives J~,ε(Φ~(ξ)) ≤

~N (m0 + k∗) for any ξ ∈ M thus Φ~(M) ⊂ Σ~,ε . Furthermore, Lemma

2.8.1 implies β(u) ∈ Mδ for any u ∈ Σ~,ε and β(Σ~,ε) ⊂ Mδ. Finally, as aconsequence of (2.8.1), the map β Φ~ is homotopically equivalent to the em-bedding j : M → Mδ in Mδ. Lemma 2.3.2 implies cat

Σ~,ε(Σ~,ε) ≥ catMδ

(M).

Therefore Theorem 2.3.1 implies that J~,ε has at least catMδ(M) critical points

on Σ~,ε.

Part II

Blow-up solutions for

semilinear problems

Chapter 3

Singular quasilinear equations

with natural growth and

semilinear problems with

blow-up at the boundary

3.1 Electrohydrodynamics. The equilibrium of a

charged gas in a container

In 1954, J.B. Keller [68] and R. Osserman [83] established the existence of asolution of the nonlinear equation

∆u = f(u),

in any bounded domain of RN with N ≥ 3. Such a solution, which becomes

infinite everywhere on the boundary of the domain provided that f(u) is anincreasing function, is called a large solution. In particular, Keller and Osser-man showed that a necessary and sufficient condition for the existence of largesolutions in bounded domains is that the function f satisfies

∫ ∞

1

[∫ s

0f(t) dt

]−1/2

ds <∞.

In [67], Keller applied such results to electrohydrodynamics, namely to theproblem of the equilibrium of a uniformly charged gas in a perfectly conduct-ing container.

96 Chapter 3. Singular quasilinear equations and blow-up semilinear problems

Equilibrium occurs when the electric forces in the gas just balance the pressureforces. It is found that for any container and for each total mass of gas thereis exactly one equilibrium distribution of that mass of gas in the container.In equilibrium the density and pressure attain their maxima at the containersurface, and they are both constant on this surface. Furthermore, the densityand pressure increase at each point as the total mass of gas increases. How-ever, inside the container the density and pressure do not increase indefinitelyas the total mass of gas does. Instead at each inner point the density andpressure both approach finite upper limits as the total mass of gas becomesinfinite. Most of the gas accumulates in a thin layer near the surface whenthe total mass is large. The fact that the density and pressure cannot bemade arbitrarily large at inner points by putting more gas into the containeris considered to be the main physically interesting result.

By ’electrohydrodynamics’ we mean the science of the motions of fluids underthe influence of electric fields. Such motions occur in vacuum tubes, wherethe fluid is an electron gas, in plasmas where the fluid is a neutral mixture ofions, in the ionosphere, etc. Whereas most previous studies of these motionshave been based upon kinetic theory, the present investigation is completelymacroscopic. The fluid is treated as a continuum, and this makes it possibleto deal with more general fluids and to obtain more complete results.In these Sections we consider the equilibrium of a massM of uniformly chargedgas (e.g., an electron gas) within a container. If the gas had no pressure, weshould conclude that all the charge would come to rest on the container wall,due to the mutual repulsion of the charges constituting the gas. However whenwe take account of the pressure we find that the gas is distributed throughoutthe container, with its greatest density at the wall. Furthermore, the densityhas the same value at all points of the wall. As M increases, the density atevery point within the container also increases.Although the preceding results are physically obvious, the main result byKeller, which we recall in the following, is not obvious. It is that the densitydoes not increase indefinitely with M at interior points. Instead, at eachinner point the density has an upper bound which depends upon the shapeof the container and the location of the point. As the point approaches theboundary, the bound increases indefinitely. Thus as M increases, most of thegas accumulates in a thin layer near the surface. The fact that the densitycannot be made arbitrarily large at points inside a container by putting moregas into the container may be of some practical importance.

3.2. Formulation of the problem 97

We also find that there exists a solution in which M is infinite but the densityat each interior point is finite. This generalizes previous results of Bieberbach[29] and Rademacher [90], concerning a charged ideal gas. Finally we presentsome additional results for an ideal gas and discuss the work of Von Laue [104].

3.2 Formulation of the problem

We consider the equilibrium of a charged fluid within a container. Such afluid achieves equilibrium when the pressure forces and electrostatic forces init balance each other. In terms of the pressure p, the mass density ρ, thecharge density aρ and the electric field vector E, this equilibrium condition is

∇p = aρE. (3.2.1)

The constant a in (3.2.1) is the ratio of electric density to mass density in thefluid. It is constant because we assume that the fluid is uniformly charged. Ifeach molecule of the fluid has mass m and charge e then a = em−1, which isthe case when the fluid is an electron gas.The fact that the charge is a source of the field is expressed by the equation

∇ · E = 4πaρ. (3.2.2)

If the container surface S is a perfect conductor then

Etangential = 0 on S. (3.2.3)

The fluid is characterized by an equation of state which expresses p as afunction of ρ and the temperature. When the temperature is constant, as weassume it to be, the equation of state takes the form

p = p(ρ). (3.2.4)

The function p(ρ) is a non-negative increasing function defined for all non-negative ρ, and becomes infinite as ρ does. The total mass of fluid within thecontainer D is given by

M =

∫

Dρ dV. (3.2.5)

Physically it seems that the preceding conditions are the only ones which mustbe satisfied by a charged gas in a perfectly conducting container. If this is so,then equations (3.2.1)-(3.2.5) should have one and only one solution p, ρ, andE for each value of the total mass M . In the next Section we shall show thatthis is the case, and we shall also determine various properties of the solution.


3.3 Existence, uniqueness and monotonicity of the

solution

By eliminating E from (3.2.1) and (3.2.2) we obtain

∇ · (ρ−1∇p) = 4πa2ρ. (3.3.1)

Since (3.2.4) expresses p as a function of ρ, (3.3.1) is an equation involving ρalone. Alternatively, since p(ρ) is monotonic, ρ may be expressed in terms ofp by means of (3.2.4) and then (3.3.1) becomes an equation for p. It is moreconvenient, however, to introduce instead a new function v defined by

v =

∫ p

p0

dp

ρ(p). (3.3.2)

Here p0 is an arbitrary constant greater than or equal to p(0). In terms of v,(3.3.1) becomes

∇2v = f(v). (3.3.3)

The function f(v) is defined in terms of the inverse functions ρ(p) and p(v) by

f(v) = 4πa2ρ[p(v)]. (3.3.4)

The function f(v) is a non-negative increasing function of v.From (3.2.3) and (3.2.1) we see that ∇p is normal to the container surfaceS, which is therefore a level surface of p. Then from (3.2.4) and (3.3.2) weconclude that p, ρ and v are constant on S if S is connected, or on eachconnected component of S if S is connected. Assuming for simplicity that S isconnected we find that v is a solution of (3.3.3) which is constant on S. Now letα denote the value of v on S. Fron the existence, uniqueness and monotonicitytheorems for solutions of (3.3.3) when f is increasing we have the followingresult: there is one and only one solution v of (3.3.3) having the value α onthe surface S of the container. This solution increases at every point as αincreases. Furthermore, from the fact that f is non-negative it follows thatv is a subharmonic function. Therefore v ≤ α throughout the container, bythe maximum principle for subharmonic functions. Corresponding results alsoapply to p and ρ since they are increasing functions of v.In the Appendix of [67], it is shown that α can be uniquely chosen so that(3.2.5) is satisfied for any given non-negative number M and that α is an

3.4. Bounds on the solution 99

increasing function of M . It then follows that equations (3.2.1)-(3.2.5) haveone and only one solution for any M ≥ 0. In this solution ρ > 0 and p > p(0)if M > 0. Both ρ and p are constant on S, attain their maximum valueson S and are increasing functions of M at each point of D. Correspondingresults can be obtained for two dimensional containers by slightly modifyingthe foregoing proofs. For a one-dimensional container (i.e., the region betweentwo parallel planes) the boundary is not connected, but the same results areobtained if it is assumed that ρ has the same value on the two planes.

3.4 Bounds on the solution

We have just seen that ρ and p increase with M . We shall now show that if∫ ∞

p(1)

dp

ρ(p)(p− p0)12

<∞ (3.4.1)

then there exists a function g(R) determined solely by ρ(p) such that

v(p) ≤ g[R(P )]. (3.4.2)

The function g(R) is decreasing and g(R) → ∞ as R→ 0. In (3.4.2) P denotesa point in D and R(P ) denotes the distance from P to S. Equation (3.4.2)shows that v(P ), and therefore also ρ and p, are bounded above independentlyof M for any gas satisfying (3.4.1). Thus v, ρ and p tend to finite upper limitsas M becomes infinite, for all points P in D. We note that (3.4.1) dependsonly upon the behavior of ρ(p) as p→ ∞ and it is satisfied if ρ increases faster

that p12 . For a polytropic gas with adiabatic exponent γ, ρ is proportional to

p1/γ so (3.4.1) holds for γ < 2.Before proving (3.4.1) we observe that the upper bound on ρ(p) (and on v andp) tends to infinity as P tends to S. Thus as M increases ρ tends to a finiteupper limit at each inner point of D and ρ becomes infinite on S. Therefore,most of the gas accumulates in a thin layer near S as M increases. There isalso a solution of (3.2.1) to (3.2.5) with M infinite, provided (3.4.1) holds. Inthis solution v, ρ and p are finite at each inner point of D but infinite on S.This follows from a result in Keller [68].Equation (3.4.2) also follows from a theorem in [68] on solutions of (3.3.3)when f(v) is positive and increasing, and for which

∫ ∞

1

[∫ x

0f(z) dz

]− 12

dx <∞. (3.4.3)


Now f(v) = ρ is positive when M > 0, and f is increasing. Therefore thetheorem applies when (3.4.3) holds, and (3.4.3) becomes (3.4.1) if the functionρ(p) inverse to (3.2.4) is introduced. A method of determining the bound g(R)is given in [68] and will not be described here.

3.5 Example: the ideal gas

To exemplify the preceding considerations, let us consider an ideal gas, forwhich the equation of state (3.2.4) is

p =RT

mρ. (3.5.1)

In (3.5.1) T is the constant temperature, R is the gas constant, and m is theaverage mass of the molecules in the gas. Then v becomes, if p0 = 1,

v =RT

mlog p. (3.5.2)

From (3.3.4), (3.5.1) and (3.5.2), we see that f(v) is now

f(v) =4πa2m

RTexp

[ mRT

v]. (3.5.3)

Then (3.3.3) becomes

∇2v =4πa2m

RTexp

[ mRT

v]. (3.5.4)

If we define a new variable u by

u =m

RTv + log 4π

[amRT

]2(3.5.5)

then (3.5.4) becomes

∇2u = eu. (3.5.6)

If the container is a sphere, a cylinder, or a pair of parallel planes, we mayassume that the solution u of (3.5.6) is a function of one variable only. Thisvariable, which we will denote by r, is the distance from the center of thesphere, from the axis of the cylinder, or from the median plane in the three-,

3.5. Example: the ideal gas 101

two-, and one-dimensional cases respectively. For details, we refer to Section5 in [68].Equation (3.5.6) was studied by M. Von Laue [104] in connection with theequilibrium of an electron gas. He deduced (3.5.6) by using the statistical me-chanical result that in equilibrium the density at any point is proportional toan exponential function in which the exponent is the negative of the potentialenergy function at the point divided by RT . The potential was then assumedto be just the electrostatic potential, which limited the considerations to anideal gas.Laue considered the problem of solving (3.5.6) for the potential u in a half-space, where u is a given constant on the boundary plane and a linear functionof distance from the plane at infinity. He also treated the corresponding prob-lem for the exterior of a circle, with appropriate conditions on u at infinity. Inboth these problems (3.5.6) becomes an ordinary differential equation whichcan be solved explicitly. In addition, he observed that by means of conformalmapping, the corresponding problem for the exterior of any cylinder couldbe obtained from that for a circular cylinder if ρ = ∞ (i.e., u = ∞) on thecylinder. From previous results we see that this solution provides an upperbound for any solution with ρ finite on the cylinder.L. Bieberbach [29] proved that in the two-dimensional case within a smoothclosed curve, (3.5.6) has a unique solution which becomes infinite to the sameorder as s−2 as the boundary curve is approached, where s denotes the dis-tance of a point from the boundary curve. H. Rademacher [90] extendedBieberbach’s result to three dimensions. Both of these results -excepting theuniqueness- are contained in Section 5 in [68], which applies to any gas forwhich (3.4.1) is satisfied, and not merely to an ideal gas.

3.6 A quasilinear approach for semilinear problems

with blow-up at the boundary

These Sections are motivated by the study of the existence of a positive solu-tion for the problem

∆u = a(x)f(u), x ∈ Ωu = ∞, x ∈ ∂Ω

(3.6.1)

where Ω is a bounded domain in RN (N ≥ 3), a ∈ Lq(Ω) with q > N/2, and

f ∈ C1[0,+∞) is a strictly positive function in (0,+∞).

This problem appears in the study of the subsonic motion of a gas [86], theelectric potential in a glowing hollow metal body [90], the Riemannian surfacesof constant negative curvature [29, 37, 79] and automorphic functions [29]. Theexistence of solutions of this problem, usually known as large solutions, hasbeen studied by many authors [16, 38, 54, 68, 71, 72, 79, 83, 86, 88, 102, 103]for smooth bounded domains. The basic idea in these works is to approximatethe solution u by a sequence un of solutions of the b.v.p.

∆un = a(x)f(un), x ∈ Ωun = n, x ∈ ∂Ω.

Observe that the boundary condition satisfied by un implies that the sequenceun is unbounded in L∞(Ω) and it is not easy to prove its convergence.Indeed, in these works it is essential to assume that f is nondecreasing toshow that un is increasing and converging to a solution u of (3.6.1).

One of our purposes in these Sections is to give sufficient conditions thatdoes not involve the monotony assumption on f for the existence of solution of(3.6.1) in general bounded, not necessarily smooth, domains. In contrast withthe previous works, where the sub- and super-solution method is essentiallyused, we develop a different approach to solve the problem (3.6.1). The basichypothesis is that there exists τ > 0 such that

+∞∫

τ

ds

f(s)<

+∞∫

0

ds

f(s)= +∞. (3.6.2)

This condition allows us to consider the real function H(u) =∫ +∞u ds/f(s),

for every u > 0 and to give our notion of solution of (3.6.1) without requiring

3.6. A quasilinear approach for blow-up semilinear problems 103

the smoothness of the boundary of the domain Ω. Indeed, for a solution wemean u ∈ H1

loc(Ω) satisfying

−

∫

Ω

∇u(x) · ∇ϕ(x) =

∫

Ω

a(x)f(u(x))ϕ(x), ∀ϕ ∈ C∞0 (Ω)

and

H(u) ∈ H10 (Ω).

Now, the change of variable H(u) = w leads us to consider the equivalentquasilinear elliptic boundary value problem

−∆w + g(x,w)|∇w|2 = a(x), x ∈ Ωw ∈ H1

0 (Ω),(3.6.3)

where g(x, s) = g(s) = f ′(H−1(s)), for every s > 0 and x ∈ Ω. Although theequation in (3.6.3) is more complicated than the one in (3.6.1), the advantageof studying (3.6.3) instead of (3.6.1) is based on the deduction of a L∞(Ω) apriori boundedness of the solutions of (3.6.3) from the inequality q > N/2.

Quasilinear equations with quadratic growth in ∇w and a Caratheodoryfunction g(x, s) in Ω × R like those in (3.6.3) have been considered in [1, 23,24, 30, 31, 32, 33, 56, 61, 87]. In particular, in [23, 30, 31, 32] it is not assumedany limitation on the growth of g(x, s) as s tends to infinity and general termsa(x) are considered. Recently, the case of a positive nonlinearity g(x, s) inΩ × (0,+∞) with a singularity at s = 0 has been studied in [13] (see also[12]). However, the application of the result in that work on the existence of apositive solution of problem (3.6.3) (for g > 0) to obtain a solution of (3.6.1)requires that f be a nondecreasing function. To overcome this requirement westrongly need to improve the cited result in [13] to handle a not necessarilynonnegative g. This is our main purpose.

It is worth while to remark that even in the previous studies [23, 30, 31, 32]of (3.6.3) for a Caratheodory (nonsingular) g : Ω×R −→ R, it is assumed thesign condition

sg(x, s) ≥ 0,

for every s ∈ R and a.e. x ∈ Ω. Indeed, this is explicitly imposed in [23, 31, 32].In [30], the authors consider a general Caratheodory term g(x,w,∇w) insteadof the pure quadratic term g(x,w)|∇w|2, and prove the existence of a solutionprovided that the nonlinearity g satisfies, instead of the sign condition, amore general “one-side condition” (see formula (2.10) in [30]). We point out


that it is easily verified that in the case of a pure quadratic term, this one-sidecondition is not more than the above sign condition. In this way, the extensionof [13] to a not necessarily nonnegative g in (0,+∞) that we are looking for,constitutes also an improvement of the earlier works [23, 30, 31, 32]. This iscarried out in the following Theorem.

Theorem 3.6.1. Consider for q > N/2 a function a ∈ Lq(Ω) satisfying

infa(x) |x ∈ ω > 0, ∀ω ⊂⊂ Ω. (3.6.4)

Let also g : Ω × (0,+∞) −→ R be a Caratheodory function. If there exist anincreasing function b : (0,+∞) −→ (0,+∞) and µ ∈ (0, 1) such that

−µ ≤ sg(x, s) ≤ b(s), ∀s > 0, a.e. x ∈ Ω, (3.6.5)

then problem (3.6.3) has at least one positive solution w ∈ H10 (Ω) ∩ L∞(Ω).

The meaning of hypothesis (3.6.5) is, roughly speaking, that g(x, s) is belowthe positive hyperbola C/s for bounded sets of values s > 0 and above from thenegative hyperbola −µ/s for every s > 0. Since b may be any nondecreasingfunction, we remark that no condition on the growth of g(x, s) as s goes toinfinity is imposed. In particular, the nonlinearity can have a singularity atzero.

To prove Theorem 3.6.1, we construct a suitable sequence of approximatedCaratheodory functions gn : Ω × R −→ R of the function g and considerthe sequence of approximated problems (problem (3.6.3) with g replaced bygn). Applying the Schauder fixed point theorem we prove the existence of asolution wn ∈ H1

0 (Ω) of this approximated problems. Then, following [23], weconclude the proof showing that wn converges to a positive solution of (3.6.3).However, the possible singularity of g(x, s) at s = 0 implies that gn(x,wn(x))can blow up as wn(x) is converging to zero. Therefore, the key-point consistsin establishing first that wn are uniformly away from zero in every compact setin Ω. Next, using a quadratic exponential test function (see [23]), we obtainthe convergence of wn in H1

loc(Ω) to a positive solution of (3.6.3).As a consequence of the previous result, we deduce the existence of a

solution u to the problem (3.6.1) by the following Theorem .

Theorem 3.6.2. Assume that f ∈ C1[0,+∞) is a positive function satisfying(3.6.2) and hypothesis (3.6.4) holds with a ∈ Lq(Ω) for some q > N/2. If thereexists µ ∈ (0, 1) such that

f ′(t)

+∞∫

t

ds

f(s)≥ −µ, ∀t > 0 (3.6.6)

3.7. The approximated quasilinear problems 105

and

lim inft→+∞

f ′(t)

+∞∫

t

ds

f(s)< +∞, (3.6.7)

then (3.6.1) has at least one positive solution u ∈ H1loc

(Ω).

The simplest example of nonlinearity f satisfying the hypothesis of theprevious theorem is f(s) = g(s)sp with p > 1 and g a continuous functionsatisfying lim infs→+∞ sg′(s) < +∞, 0 < k1 ≤ g(s) ≤ k2 < +∞ and pg(s) +sg′(s) > −k1(p − 1), for every s > 0. Such results are contained in a recentpaper with D. Arcoya and P.J. Martınez-Aparicio [11].

From the pioneering works [68, 83] where it is proved that, if a(x) ≡ 1, then(3.6.1) has a solution if and only if f satisfies the Keller-Osserman condition(see Remark 3.7.8), several improvements have been done. In particular, in [71]it is shown the necessity and sufficiency of the Keller-Osserman condition inthe case of a general nonnegative and continuous function a(x) and a positiveand nondecreasing function f with f(0) = 0. More recently, in [54] it isconsidered the case of a not necessarily monotone f and a sharpened Keller-Osserman condition is given. In such paper, the authors use a well-knownresult by Gidas, Ni and Nirenberg in [58] and require that a(x) ≡ 1. On theother hand, other results have been obtained in [38, 102] by assuming someconditions on the behavior of a(x) near the boundary.

3.7 The approximated quasilinear problems

We devote this Section to prove Theorem 3.6.1. Denote by S the Sobolevconstant, i.e.

S = infu∈H1

0 (Ω)\0

‖u‖2

‖u‖22∗,

with 2∗ = 2N/(N − 2). Consider also

C = (1 − µ)−2∗ S−2∗‖a‖2∗

q |Ω|β−122∗ β

β−1 , (3.7.1)

where β ≡ 2∗[

1(2∗)′ −

1q

]. By (3.6.5), there exists Λ > 1 such that

sg(x, s) ≤ Λ, ∀s ∈ (0, C]. (3.7.2)


We are going to approximate (3.6.3) by a sequence of problems. Indeed, if wedenote by Tk the truncature function given by

Tk(s) =

−k if s ≤ −k,s if − k < s < k,k if k ≤ s,

(3.7.3)

we define the Caratheodory function gn in Ω × R by

gn(x, s) =

0 if s ≤ 0,n2s2Tn(g(x, s)) if 0 < s < 1

n ,Tn(g(x, s)) if 1

n ≤ s.

It is easy to verify that gn is bounded and satisfies

gn(x, s)(n→+∞)−→ g(x, s), ∀x ∈ Ω, ∀s > 0.

Furthermore, by (3.7.2),

gn(x, s) ≤Λ

s, ∀s ∈ (0, C] a.e. x ∈ Ω, (3.7.4)

and, by (3.6.5),

sgn(x, s) + µ ≥ 0 a.e. x ∈ Ω, ∀s ∈ R, (3.7.5)

for every n ∈ N.We claim that there exists at least one solution wn ∈ H1

0 (Ω) of the approx-imated problem

−∆w + gn(x,w)|∇w|2

1 + 1n |∇w|

2= a(x) in Ω

w ∈ H10 (Ω).

(3.7.6)

To prove it, we recall that a ∈ Lq(Ω) with q > N2 and define the operator

Sn : H10 (Ω) −→ Lq(Ω) by

Sn(w) = −gn(x,w)|∇w|2

1 + 1n |∇w|

2+ a(x).

As an application of the dominated convergence theorem, if wk → w in H10 (Ω)

we infer, from the convergences wk −→ w and ∇wk −→ ∇w a.e. x ∈ Ω, the


boundedness of the functions gn(x, s) (x ∈ Ω, s ∈ R) and |ξ|2/(1 + |ξ|2/n),(ξ ∈ R

N ) that

∫

Ω

∣∣∣∣∣

[gn(x,wk)

|∇wk|2

1 + 1n |∇wk|

2− gn(x,w)

|∇w|2

1 + 1n |∇w|

2

]∣∣∣∣∣

q(k→+∞)−→ 0,

i.e., Snwk converges to Snw in Lq(Ω) as k tends to infinity. Therefore, Sn iscontinuous.

Considering now the inverse of the Laplacian operator (−∆)−1 : Lq(Ω) −→H1

0 (Ω) we see that the solutions of (3.7.6) are just the fixed points of thecomposition operator Qn ≡ (−∆)−1 Sn. Since q > N/2 we deduce thatthe operator (−∆)−1 is compact and hence the composition of it with thecontinuous operator Sn, i.e. Qn, is also compact.

On the other hand, since |gn(x, s)| ≤ n and |ξ|2/(1 + |ξ|2/n)| ≤ n, weobserve that for every w ∈ H1

0 (Ω)

‖Snw‖q ≤

∥∥∥∥∥−gn(x,w)|∇w|2

1 + 1n |∇w|

2

∥∥∥∥∥q

+ ‖a‖q ≤ n2|Ω| + ‖a‖q,

which by the continuity of (−∆)−1 implies that there exists R > 0 such that

‖Qnw‖H10 (Ω) < R

for every w ∈ H10 (Ω). In particular, the compact operator Qn maps the

ball in H10 (Ω) centered at zero with radius R into itself. The Schauder fixed

point theorem shows that there exists a fixed point wn ∈ H10 (Ω) of Qn, or

equivalently, a solution of (3.7.6).

Now, we are going to prove that wn are a priori bounded in both H10 (Ω)

and L∞(Ω) spaces. Indeed, if we take ϕ = wn as test function in (3.7.6) itfollows that

∫

Ω|∇wn|

2 +

∫

Ωgn(x,wn)

|∇wn|2

1 + 1n |∇wn|

2wn =

∫

Ωa(x)wn,

or, equivalently,

(1 − µ)

∫

Ω|∇wn|

2

+

∫

Ω

[gn(x,wn)

|∇wn|2

1 + 1n |∇wn|

2wn + µ|∇wn|

2

]=

∫

Ωa(x)wn.


Observing that, by (3.7.5), sgn(x, s)|ξ|2/(1 + 1

n |ξ|2) + µ|ξ|2/(1 + 1

n |ξ|2) ≥ 0,

a.e. x ∈ Ω, for every s ∈ R, ξ ∈ RN and hence that

sgn(x, s)|ξ|2

1 + 1n |ξ|

2+ µ|ξ|2 ≥ 0, (3.7.7)

we get

(1 − µ) ‖wn‖2H1

0≤

∫

Ωa(x)wn ≤ ‖a‖q‖wn‖q′ ,

where q′ is the conjugate exponent of q, i.e. q′ = q/(q − 1). Since q > N/2we have q′ < 2∗ and hence by Sobolev embedding theorem it is clear that thesequence wn is bounded in H1

0 (Ω).

To prove the a priori estimate in L∞(Ω), we define the truncature functionGk(s) = s − Tk(s), where Tk is given by (3.7.3), and we take ϕ = Gk(wn) astest function in (3.7.6) to obtain

(1 − µ)

∫

Ω|∇Gk(wn)|

2

+

∫

Ω

[gn(x,wn)

|∇wn|2

1 + 1n |∇wn|

2Gk(wn) + µ|∇Gk(wn)|

2

]=

∫

Ωa(x)Gk(wn).

Using (3.7.7) we deduce that

(1 − µ)

∫

Ω|∇Gk(wn)|

2 ≤

∫

Ωa(x)Gk(wn).

Since µ < 1, by the method of Stampacchia (see [97]) it follows, from thisinequality, that

‖wn‖∞ ≤ (1 − µ)−2∗ S−2∗‖a‖2∗

q |Ω|β−122∗ β

β−1 = C.

On the other hand, taking w−n ≡ minwn, 0 as test function in (3.7.6) we

obtain

∫

Ω|∇w−

n |2 +

∫

Ωgn(x,wn)

|∇wn|2

1 + 1n |∇wn|

2w−n =

∫

Ωa(x)w−

n ,


that is,

(1 − µ)

∫

Ω|∇w−

n |2

+

∫

Ω

[gn(x,wn)

|∇wn|2

1 + 1n |∇wn|

2w−n + µ|∇w−

n |2

]=

∫

Ωa(x)w−

n

and by (3.6.4) and (3.7.7) we get

(1 − µ)

∫

Ω|∇w−

n |2 ≤

∫

Ωa(x)w−

n ≤ 0.

Therefore, since µ < 1 we deduce that wn ≥ 0.In addition, by following [12] we prove that wn > 0 in Ω. Indeed, taking

Cn > 0 such that gn(x, s) ≤ Cns, for s ∈ [0, C] (where C is given by (3.7.1)),we have

gn(x,wn(x))|∇wn(x)|

2

1 + 1n |∇wn(x)|

2≤ nCnwn(x), ∀x ∈ Ω.

Consequently,

−∆wn + nCnwn ≥ −∆wn + gn(x,wn)|∇wn|

2

1 + 1n |∇wn|

2= a(x) ≥ 0, x ∈ Ω,

wn(x) ≥ 0, a.e. x ∈ Ω.

By hypothesis (3.6.4), a 6≡ 0 in every ball B ⊂ Ω and hence the strong maxi-mum principle (see [60]) implies wn > 0 in B for every ball B ⊂ Ω. Thereforewn > 0 in Ω.

Now, we are going to prove that the sequence of the approximated solutionsis uniformly away from zero in every compact set in Ω.

Lemma 3.7.1. For every Ω0 ⊂⊂ Ω there exists a constant cΩ0 > 0 such that

wn(x) ≥ cΩ0 , a.e. x ∈ Ω0.

Proof. Since wn > 0, inequality (3.7.4) implies that wn ∈ H10 (Ω) satisfies, in

the weak sense,

a(x) = −wn + gn(x,wn)|∇wn|

2

1 + 1n |∇wn|

2≤ −wn +

Λ

wn|∇wn|

2, x ∈ Ω


i.e., wn is a supersolution of

−w + Λw |∇w|

2 = a(x), x ∈ Ω

w ∈ H10 (Ω).

Hence we infer that un = (Λ − 1)Λ−1w1−Λn is a subsolution of

∆u(x) = a(x)u(x)Λ

Λ−1 , x ∈ Ω,

and we can apply the following result (see Proposition 2.3 in [13]).

Proposition 3.7.2. Let h : [0,+∞) −→ [0,+∞) be a nondecreasing continu-ous function such that h(t) > 0 for t > 0 and

+∞∫

a

dx√x∫ah(z) dz

< +∞, ∀a > 0. (3.7.8)

Assume also that a(x) ∈ Lq(Ω), q > N/2, satisfies (3.6.4). Then for everyopen set Ω0 ⊂⊂ Ω there exists a positive constant CΩ0 such that for everyu ∈ L∞(Ω) ∩H1(Ω) satisfying

∆u ≥ a(x)h(u), x ∈ Ω

it holdsu(x) ≤ CΩ0 , a.e. x ∈ Ω0.

Remark 3.7.3. The hypothesis (3.7.8) is usually called the Keller-Ossermancondition and was introduced in [68, 83] for the study of problem (3.6.1)

Proof of Lemma 3.7.1 completed. Fix now Ω0 ⊂⊂ Ω. By Proposition 3.7.2

with h(s) = sΛ

Λ−1 , there exists CΩ0 > 0 such that

un ≤ CΩ0 , a.e. x ∈ Ω0

and thus

wn = (Λ − 1)u−1/(Λ−1)n ≥ (Λ − 1)C

−1/(Λ−1)Ω0

≡ cΩ0 > 0, a.e. x ∈ Ω0.

3.8. Positive solutions for the quasilinear problem 111

3.8 Positive solutions for the quasilinear problem

In this Section we study the convergence of the approximated solutions wnto a solution w of (3.6.3). Since wn is bounded in H1

0 (Ω) and in L∞(Ω), wecan assume that there exists w ∈ H1

0 (Ω) ∩ L∞(Ω) such that the followingconvergences hold:

wnw, (weakly) in H10 (Ω),

wn(x) → w(x), a.e. x ∈ Ω.(3.8.1)

In the following Lemma, we are going to prove that for every ω ⊂⊂ Ω, wnconverges to w in H1(ω). The proof is based on a suitable choice of a test func-tion of quadratic exponential type [31] in conjunction with the Lemma 3.7.1.

Lemma 3.8.1. For every ψ ∈ C∞0 (Ω) such that ψ ≥ 0, we have

limn→+∞

∫

Ω

|∇(wn − w)|2ψ(x)dx = 0.

Proof. Let ψ be a function in C∞0 (Ω) such that ψ ≥ 0. Choose Ω0 ⊂⊂ Ω

such that suppψ ⊂ Ω0. From Lemma 3.7.1 there exists c0 > 0 such thatwn(x) ≥ c0 > 0, a.e. x ∈ Ω0.

From (3.7.4) and (3.7.5) we can consider c = supgn(x, s) | x ∈ Ω, c0 ≤s ≤ C and ϕγ(s) = seγs

2, where γ is large enough to satisfy

ϕ′γ(s) − c|ϕγ(s)| = eγs

2 [1 + 2γs2 − c|s|

]≥

1

2, ∀s ∈ R. (3.8.2)

Take ϕγ(wn − w)ψ(x) as test function in (3.7.6) and obtain

∫

Ω0

∇wn∇(wn − w)ϕ′γ(wn − w)ψ +

∫

Ω0

∇wn∇ψϕγ(wn − w) +

+

∫

Ω0

gn(x,wn)|∇wn|

2

1 + 1n |∇wn|

2ϕγ(wn − w)ψ =

∫

Ω0

a(x)ϕγ(wn − w)ψ.


Adding and subtracting∫Ω0

∇w∇(wn − w)ϕ′γ(wn − w)ψ,

∫

Ω0

|∇(wn − w)|2ϕ′γ(wn − w)ψ +

∫

Ω0

∇w∇(wn − w)ϕ′γ(wn − w)ψ

+

∫

Ω0


∫

Ω0

gn(x,wn)|∇wn|

2

1 + 1n |∇wn|

2ϕγ(wn − w)ψ

=

∫

Ω0

aϕγ(wn − w)ψ.

Observing that

gn(x,wn)|∇wn|

2

1 + 1n |∇wn|

2ϕγ(wn−w)ψ ≥ −c|∇wn|

2|ϕγ(wn−w)|ψ, a.e. x ∈ Ω0,

we derive that

∫

Ω0

|∇(wn − w)|2ϕ′γ(wn − w)ψ − c

∫

Ω0

|∇wn|2|ϕγ(wn − w)|ψ

≤ −

∫

Ω0

∇w∇(wn − w)ϕ′γ(wn − w)ψ −

∫

Ω0

∇wn∇ψϕγ(wn − w)

+

∫

Ω0

aϕγ(wn − w)ψ.

Observing that

∫

Ω0

|∇(wn − w)|2|ϕγ(wn − w)|ψ =

∫

wn−w<0

|∇wn|2|ϕγ(wn − w)|ψ

+

∫

wn−w<0

|∇w|2|ϕγ(wn − w)|ψ − 2

∫

wn−w<0

∇wn∇w|ϕγ(wn − w)|ψ,

we deduce from (3.8.2) that

3.8. Positive solutions for the quasilinear problem 113

1

2

∫

Ω0

|∇(wn − w)|2ψ ≤ −c

∫

wn−w<0

|∇w|2|ϕγ(wn − w)|ψ

+2c

∫

wn−w<0

∇wn∇w|ϕγ(wn − w)|ψ −

∫

Ω0

∇w∇(wn − w)ϕ′γ(wn − w)ψ

−

∫

Ω0


∫

Ω0

aϕγ(wn − w)ψ.

By (3.8.1), we can use the dominated convergence theorem to see that each ofthe terms in the right hand of this inequality converges to zero thus showingthat

limn→+∞

∫

Ω0

|∇(wn − w)|2ψ = 0.

We conclude this Section by showing that the limit w of the approximatedsolutions wn is a solution of (3.6.3), i.e., proving our main result for suchquasilinear problem.

Proof of Theorem 3.6.1 completed. By Lemma 3.8.1, for every ω ⊂⊂ Ω, thereexists hω ∈ L2(ω) such that, up to a subsequence,

|∇wn(x)| ≤ hω(x), a.e. x ∈ ω, (3.8.3)

∇wn(x) → ∇w(x), a.e. x ∈ ω. (3.8.4)

In particular, we have

∇wn(x) → ∇w(x) a.e. x ∈ Ω.

Now, we can pass to the limit in the equation satisfied by the approximatedsolutions wn, i.e. in the integral identities

∫

Ω

∇wn∇φ+

∫

Ω

gn(x,wn)|∇wn|2φ =

∫

Ω

aφ, ∀φ ∈ C∞0 (Ω).

Indeed, if φ ∈ C∞0 (Ω) then the weak convergence of wn to w implies that

limn→+∞

∫

Ω

∇wn∇φ =

∫

Ω

∇w∇φ.


On the other hand, from Lemma 3.7.1, there exists c0 > 0 such that wn(x) ≥c0 > 0, a.e. x ∈ ω ≡ suppφ. Thus, by (3.7.4) and (3.7.5), we conclude thatfor some c > 0 we have |gn(x,wn(x))| ≤ c, a.e. x ∈ ω. Therefore, by (3.8.3),

|gn(x,wn(x))||∇wn(x)|2

1 + 1n |∇wn(x)|

2≤ ch2

ω(x) a.e. x ∈ ω,

with h2ω ∈ L1(ω). In addition, by the definition of gn and (3.8.4), for n > 1/c0

we have gn(x,wn(x)) = Tn (g(x,wn(x))) and thus

gn(x,wn(x))|∇wn(x)|

2

1 + 1n |∇wn(x)|

2−→ g(x,w(x))|∇w(x)|2 a.e. x ∈ ω.

By the Lebesgue dominated convergence Theorem we show that

limn→+∞

∫

Ω

gn(x,wn)|∇wn|

2

1 + 1n |∇wn|

2φ =

∫

Ω

g(x,w)|∇w|2φ.

Therefore, passing to the limit as n goes to infinity in the equation satisfiedby wn we deduce that

∫

Ω

∇w∇φ+

∫

Ω

g(x,w)|∇w|2φ =

∫

Ω

aφ, ∀φ ∈ C∞0 (Ω),

i.e. w ∈ H10 (Ω) is a solution of −∆w + g(x,w)|∇w|2 = a in Ω.

3.9 Semilinear problems with blow-up at the bound-

ary

In this Section we study problem (3.6.1). We begin by applying Theorem 3.6.1to prove Theorem 3.6.2.

Proof of Theorem 3.6.2. Let H : (0,+∞) −→ R be given by

H(t) =

+∞∫

t

ds

f(s), ∀t > 0.

3.9. Semilinear problems with blow-up at the boundary 115

We consider the b.v.p (3.6.3) where the function g is defined by g(x, s) =g(s) = f ′(H−1(s)), for every s ∈ (0,+∞). Note that by (3.6.6) with t =H−1(s),

sg(s) = sf ′(H−1(s)) = f ′(t)H(t) ≥ −µ, ∀s > 0.

On the other hand, the monotony of H implies that t = H−1(s) converges to+∞ as s goes to zero. Hence, we infer from (3.6.7) that

lim sups→0

sg(s) = lim sups→0

sf ′(H−1(s))

= lim inft→+∞

f ′(t)H(t)

= lim inft→+∞

f ′(t)

+∞∫

t

ds

f(s)< +∞,

thus concluding that (3.6.5) is satisfied with b(s) := suprg(r) | r ∈ (0, s].Therefore, from Theorem 3.6.1 we deduce that problem (3.6.3) has one

solution w ∈ H10 (Ω) ∩ L∞(Ω), i.e,

∫

Ω

∇w(x) · ∇ψ(x)dx+

∫

Ω

g(w(x))|∇w(x)|2ψ(x)dx =

∫

Ω

a(x)ψ(x)dx,

for every ψ ∈ C∞0 (Ω). In addition, for every ω ⊂⊂ Ω there exists cω > 0 such

thatcω ≤ w(x) ≤ C, a.e. x ∈ ω, (3.9.1)

with the constant C given by (3.7.1). Considering u = H−1(w) and takinginto account that

∇w = H ′(u)∇u = −1

f(u)∇u,

we get from the above that

−

∫

Ω

1

f(u(x))∇u(x) · ∇ψ(x)dx+

∫

Ω

f ′(u(x))ψ(x)|∇u(x)|2

f(u(x))2dx =

∫

Ω

a(x)ψ(x)dx,

for every ψ ∈ C∞0 (Ω). For ϕ ∈ C∞

0 (Ω) and using (3.9.1), we have ψ =(f u)ϕ ∈ H1

0 (Ω) ∩ L∞(Ω) and taking it as test function in the previousequality, we obtain

−

∫

Ω

∇u(x) · ∇ϕ(x) =

∫

Ω

a(x)f(u(x))ϕ(x), ∀ϕ ∈ C∞0 (Ω).


This means that u solves the equation

∆u = a(x)f(u), x ∈ Ω.

The proof is completed by noting that H(u) = w ∈ H10 (Ω).

2

We conclude by proving that, in some particular case, the solution u givenby Theorem 3.6.2 satisfies the boundary condition

limdist(x,∂Ω)→0

u(x) = +∞. (3.9.2)

Corollary 3.9.1. If, in addition to the hypotheses of Theorem 3.6.2, we as-sume that the boundary ∂Ω of Ω is C1,1 and f ′ ≥ 0, then the solution u givenby this Theorem satisfies (3.9.2).

Proof. Denote by v the unique solution in H10 (Ω) of −∆v = a(x) in Ω. Since

a ∈ Lq, q > N2 and ∂Ω is C1,1, by Lp-theory (see [60]) we deduce that v ∈

Cα(Ω) (α ∈ (0, 1)). In addition, using f ′ ≥ 0, we have g(w(x)) ≥ 0 for everyx ∈ Ω and

−∆w ≤ a(x), x ∈ Ω.

Thus, the comparison principle implies that 0 < w(x) ≤ v(x), for every x ∈ Ω,and by the monotony of H we get

u(x) = H−1(w(x)) ≥ H−1(v(x)), a.e. x ∈ Ω.

Finally, the continuity of v in Ω implies that limdist(x,∂Ω)→0 v(x) = 0 and,

by taking into account, in the last inequality that lims→0H−1(s) = +∞, we

conclude that (3.9.2) holds.

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Existence of solutions for some nonlinear elliptic problems · iv Index 2.9 The Palais-Smale ... This thesis deals with the study of some nonlinear elliptic problems coming from Mathematical