On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting

NoDEANonlinear differ. equ. appl. 6 (1999) 207 – 2251021-9722/99/020207-19 $ 1.50+0.20/0

c© Birkhauser Verlag, Basel, 1999

Nonlinear Differential Equationsand Applications NoDEA

On principal eigenvaluesfor quasilinear elliptic differential operators:

an Orlicz-Sobolev space setting ∗

M. GARCIA-HUIDOBRODepartamento de Matematicas, Universidad Catolica de Chile

Casilla 306, Correo 22, Santiago, Chile

V. K. LEDepartment of Mathematics & Statistics, University of Missouri-Rolla

Rolla, MO 65409, USA

R. MANASEVICHDepartamento de Ingenierıa Matematica, Universidad de Chile

Casilla 170, Correo 3, Santiago, Chile

K. SCHMITTDepartment of Mathematics, University of Utah

Salt Lake City, UT 84112, USA

1 Introduction

In this paper we consider eigenvalue problems for quasilinear elliptic partial differ-ential equations which are motivated by eigenvalue problems for the p-Laplacian(cf. [3], [8], [9], [19]). As in [10] we shall consider more general problems, which in-volve nonhomogeneous differential operators. Whereas there we considered bound-ary value problems in spaces of radial functions, (and hence obtained problems fornonlinear ordinary differential equations), here we consider the problem on generaldomains and find an Orlicz-Sobolev space ([1], [15], [17]) setting to be a suitableframework for the problem. We reduce the problem of the existence of eigenvaluesof the nonlinear elliptic equation to the question of the existence of a minimumof a coercive extended real valued functional (defined in an Orlicz-Sobolev space)which is subject to a constraint. The existence of the eigenvalue then will follow∗MG-H and RM were supported by grants CI 1∗ - CT93 - 0323 from the EC and Fondecyt

1970332-97, KS was supported by a grant from NSF, VL was supported by a grant from UMResearch Board

208 M. Garcıa-Huidobro, V. K. Le, R. Manasevich and K. Schmitt NoDEA

by an application of general properties of subdifferentials and arguments as thosepresented by Kubrusly [16] and of Liusternik’s theorem (applied in a suitable sub-space) on Lagrange multipliers ([24], [26]). This will yield what will be called aprincipal eigenvalue (for a given level). The existence of nonprincipal eigenvalueswill follow from arguments based on Liusternik-Shnirelman theory similar to thoseused in [2], [4] and [25]. Arguments based on bifurcation theory for variational in-equalities, as presented in [18], may also be used once more detailed information onthe nonlinearities is available. It will be important for our work that the subspacein which the functional acts is contained in the dual space of a separable Banachspace. This will be the case if Φ∗, the complementary function of Φ (which definesthe Orlicz-Sobolev space), satisfies a ∆2 condition. If both satisfy such a condition,then the corresponding Orlicz-Sobolev spaces are reflexive and the proofs are morestraightforward.

Thus our work may be considered to be an eigenvalue theory for rapidlygrowing operators as considered in [5], [12], and [13].

Let φ(s) = sA(s2) and ψ(s) = sB(s2) be odd increasing homeomorphisms ofR onto R and let Φ(s) =

∫ s0 φ(t) dt, Ψ(s) =

∫ s0 ψ(t) dt (such functions are called

Young or N -functions ([1], [15])).The eigenvalue problem we shall consider is the following:

(E)−div(A(|∇u|2)∇u) = λψ(u) in Ω

u = 0 on ∂Ω,

where Ω is a bounded domain in RN with a Lipschitz continuous boundary.Let Φ1 and Φ2 be two N -functions. One says that Φ1 and Φ2 are equivalent

at infinity provided there exist positive constants ci, Ti, i = 1, 2 such that

Φ1(t) ≤ Φ2(c1t), t ≥ T1, and Φ2(t) ≤ Φ1(c2t), t ≥ T2.

Let us setΦ∗(s) =

∫ s

0φ−1(τ) dτ, Ψ∗(s) =

∫ s

0ψ−1(τ) dτ.

One also says that two N -functions Φ1 and Φ2 are complementary, provided thatΦ2 = Ψ∗ for some Ψ which is equivalent to Φ1. For the N -function Φ one definesthe Sobolev conjugate Φ∗ by

Φ−1∗ (t) =

∫ t

0

Φ−1(τ)

τN+1N

dτ,

whenever the latter integral diverges as t→∞ and∫ 1

0

Φ−1(τ)

τN+1N

dτ <∞, see [1].

If this is the case, we shall assume that the N -function Ψ grows essentiallymore slowly than Φ∗ in the sense that

limt→∞

Ψ(t)Φ∗(kt)

= 0,

for all k > 0; otherwise, no other additional assumptions will be imposed on Ψ.

Vol. 6, 1999 On principal eigenvalues for quasilinear elliptic operators 209

Under such assumptions we shall show the existence of values of λ such thatproblem (E) (either in the form stated or in a weaker sense as observed below)has nontrivial positive (negative) solutions. If it is the case that Φ = Ψ (or moregenerally that Φ is equivalent to Ψ, see [15]) and Φ satisfies a ∆2 condition, wealso conclude that the set of such λ is bounded away from 0 by, say, λ1, and forany λ < λ1 problem (E) has no non-trivial solutions.

This type of problem has been studied extensively in recent years and muchinformation about eigenvalues is available in case of the p-Laplacian, i.e.

φ(s) = |s|p−2s = ψ(s), p > 1,

(see e.g. [3], [7], [8], [23]), and also in the case of general φ but Ω a ball (see[10] and the references in this paper). On a more abstract level, results abouteigenvalues of certain classes of monotone operators may be found in [2], [4], [25].The one dimensional case (i.e. the case of ordinary differential operators) has beenanalyzed most completely and we refer to [11] and its references.

In Section 4, as an application of our results, we will prove the existence ofsolutions for the problem

(N)−div(A(|∇u|2)∇u) = p(x, u) in Ω

u = 0 on ∂Ω,

where p : Ω×R→ R satisfies Caratheodory conditions and a growth condition ofthe form

|p(x, u)| ≤ aφ(u) + b(x) (1.1)

where a ∈ R is such that a < λ1 and b ∈ LΦ∗ (the Orlicz space defined by Φ∗).If it is the case that both Φ and Φ∗ satisfy a ∆2 condition at infinity, i.e.

there exists s0 > 0 such that

Φ(σs) ≤ α(σ)Φ(s), for all σ > 1, s ≥ s0, (1.2)and Φ∗(σs) ≤ β(σ)Φ∗(s), for all σ > 1, s ≥ s0, (1.3)

then more detailed information on the eigenvalues will become available. (The ∆2condition will be called global if s0 = 0).

We shall first recall some facts, essential to our work, on Orlicz-Sobolev spacesand then proceed to use a variational approach to establish the existence of eigen-values for problem (E). This will be accomplished by minimizing a coercive weak∗

lower semicontinuous convex functional on a suitable weak∗ closed set. The ex-istence of so-called principal eigenvalues will then follow using results about La-grange multipliers for the derivatives of the functionals restricted to suitable sub-spaces. Once the N -function Φ satisfies a ∆2 condition we are able to prove thatthe principal eigenvalues are bounded below and a Fredholm like result will beverified for problem (N).

The results presented here extend, to general domains, some of the resultspresented in [10] and some of the results in [20], [21] for more general nonlinearities.


2 Preliminary considerations

2.1 On Orlicz spaces

Let us recall (viz. [1], [15], [17]) that the Orlicz class LΦ is the set of all measurablesfunctions u (equivalence classes) defined on Ω such that∫

ΩΦ(|u(x)|) dx <∞,

and the Orlicz space LΦ is the linear hull of this class. This space is a Banachspace when endowed with the (Luxemburg) norm

‖u‖Φ = infk > 0 :

∫Ω

Φ(|u(x)|k

)dx ≤ 1

.

An equivalent norm, the Orlicz norm, is defined as

‖u‖(Φ) = supv∈LΦ∗ :

∫Ω Φ∗(v)≤1

∣∣∣∣∫Ωuvdx

∣∣∣∣and one has

‖u‖Φ ≤ ‖u‖(Φ) ≤ 2‖u‖Φ.

The closure in LΦ of the set of bounded functions is denoted by EΦ. This set iscomplete also and these two spaces equal if and only if a ∆2 condition holds. Inthe general case EΦ, however, is separable.

We shall also consider the Orlicz-Sobolev space W 1LΦ = W 1LΦ(Ω) definedby

W 1LΦ =u ∈ LΦ : ∂iu =

∂u

∂xi∈ LΦ, i = 1, · · · , N

.

This space is a Banach space with respect to the norm

‖u‖1,Φ = ‖u‖Φ +N∑i=1

∥∥∥∥ ∂u∂xi∥∥∥∥

Φ.

W 1EΦ is defined analogously; it again is separable. The Banach space W 10EΦ =

W 10EΦ(Ω) is the closure of C∞0 (Ω) in W 1LΦ with respect to the norm ‖ · ‖1,Φ.

We shall need embedding results and characterizations of the dual spaceswhose proofs may be found in the already cited texts [1], [14], and [17] and also inthe papers [12], [13].

Theorem 2.1 Let Φ and Φ∗ be complementary N-functions. Then the space LΦis the dual space of EΦ∗ and LΦ∗ is the dual of EΦ.


As defined above the Orlicz-Sobolev spaces W 1EΦ and W 1LΦ may be iden-tified with closed subspaces of the Cartesian product of N + 1 copies of the Orliczspaces EΦ and LΦ endowed with the product topology, denoted, respectively, byΠEΦ and ΠLΦ, i.e.

W 1LΦ =

w ∈

N∏i=0

LΦ : wi = ∂iu ∈ LΦ, 0 ≤ i ≤ N, ∂0u = u

,

with a similar identification for W 1EΦ. Then it follows from Theorem 2.1 thatW 1LΦ is a closed subspace of ΠLΦ the latter being the dual of the separable spaceΠEΦ∗ and W 1LΦ is weak∗ closed in ΠLΦ. Define W 1

0LΦ as the weak∗ closure ofC∞0 (Ω) in W 1LΦ; hence W 1

0LΦ is a weak∗ closed subset of the dual of a separablespace.

The embedding results needed are the following (see [1], [17]):

Theorem 2.2 Let Φ and Ψ be N-functions. If∫ ∞1

Φ−1(t)

tN+1N

dt =∞, (2.1)

and Ψ grow essentially more slowly than Φ∗, then the embedding

W 1LΦ → LΨ is compact and the embedding W 1LΦ → LΦ∗

is continuous. Whereas, if ∫ ∞1

Φ−1(t)

tN+1N

dt <∞,

then the embedding

W 1LΦ → LΨ is compact and the embedding W 1LΦ → L∞

is continuous.

To be able to prove the existence of eigenvalues for problem (E) in the caseΨ = Φ we need the following proposition.

Proposition 2.1 Any N-function Φ grows essentially more slowly than itsSobolev conjugate Φ∗, whenever the latter is defined as an N−function.

Proof. It follows from [1] that Φ grows essentially more slowly than Φ∗ whenever

limt→+∞

Φ−1∗ (t)

Φ−1(t)= 0.

On the other hand, for t ≥ t0

Φ−1∗ (t) = Φ−1

∗ (t0) +∫ t

t0

Φ−1(s)s−1− 1N ds,


hence, integrating by parts and then dividing by Φ−1(t), we find that

Φ−1∗ (t)

Φ−1(t)=

c1Φ−1(t)

−Nt−1/N +N

Φ−1(t)

∫ t

t0

(Φ−1)′(s)s−1/Nds,

where c1 is a constant. Using L’Hopital’s rule we get

limt→+∞

Φ−1∗ (t)

Φ−1(t)= limt→+∞

(Φ−1)′(t)t−1/N

(Φ−1)′(t)= 0.

We next state a Poincare type inequality for Orlicz-Sobolev spaces whoseproof follows immediately from a similar result in [12]. We state it as:

Lemma 2.1 Suppose that Φ is an N-function. Then∫Ω

Φ(|u|) dx ≤∫

ΩΦ(d|∇u|) dx, (2.2)

for any u ∈W 10LΦ, where d is twice the diameter of Ω.

This lemma has as a consequence:

Lemma 2.2 ‖u‖1,Φ and ‖|∇u|‖Φ define equivalent norms on W 10LΦ.

2.2 A theorem of Kubrusly

In order to prove the existence of eigenvalues of our problem we shall need ageneral result about Lagrange multipliers which is due to Kubrusly [16]. The resultis stated below and applied later (in the space U = W 1

0LΦ). We refer to the paper[16] for its proof.

Theorem 2.3 (Theorem 2, [16]) Let U be a normed linear space and F,G : U →R be two Frechet differentiable functionals whose gradients are F ′, G′ : U → U∗.Let P : U → R ∪ ∞ be a proper, convex functional with effective domain D(P ).

If F + P achieves its minimum restricted to the nonempty set

∂GR = u ∈ U : G(u) = R, R > 0,

at an element u ∈ ∂GR satisfyingv ∈ U : 〈G′(u), v − u〉 > 0 ∩D(P ) 6= ∅,v ∈ U : 〈G′(u), v − u〉 < 0 ∩D(P ) 6= ∅, (2.3)

then there exists λ ∈ R such that u is a solution of the variational inequality:

〈F ′(u)− λG′(u), v − u〉+ P (v)− P (u) ≥ 0, ∀v ∈ U. (2.4)


3 Existence of eigenvalues

In this section we shall consider the existence of eigenvalues for problem (E), i.e.the existence of values of λ such that problem (E) will have positive (or negative)solutions. To establish the existence of such eigenvalues, we shall minimize an ap-propriate functional subject to a constraint in the weak∗ closed subspace W 1

0LΦ ofthe Orlicz-Sobolev space W 1LΦ. We define functionals f, g : W 1

0LΦ → R∪∞ by

f(u) =∫

ΩΦ(|∇u|) dx, (3.1)

and g(u) =∫

ΩΨ(|u|) dx, (3.2)

and consider the minimization problem:

minu∈W 1

0LΦ:g(u)=µf(u), (3.3)

where µ is a given positive number. Note that both functionals are convex.We shall denote by

Mµ = u ∈W 10LΦ : g(u) = µ and α = inf

u∈Mµ

f(u).

3.1 On properties of the functionals

Lemma 3.1 The functional f is coercive.

Proof. Let ‖u‖ = ‖|∇u|‖Φ ≥ 1 + ε, ε > 0, then by the convexity of Φ we havethat

1 + ε

‖u‖

∫Ω

Φ(|∇u|)dx ≥∫

ΩΦ(

(1 + ε)|∇u|‖u‖

)dx > 1,

(recall the definition of the Luxemburg norm) and hence, for such u,∫Ω

Φ(|∇u|)dx > ‖u‖1 + ε

.

Hence for ‖u‖ > 1, ∫Ω

Φ(|∇u|)dx ≥ ‖u‖.

Lemma 3.2 The functional f is weak∗ lower semicontinuous.

Proof. Let un ∗ u in W 10LΦ. Theorem 2.1 then implies that (recall that W 1

0LΦis a weak∗ closed subspace of the dual of a separable space)∫

Ωunφdx→

∫Ωuφdx,

∫Ω

∂un∂xi

φdx→∫

Ω

∂u

∂xiφdx, ∀φ ∈ EΦ∗ . (3.4)


In particular, this holds for all φ ∈ L∞(Ω). Hence

∂un∂xi

∂u

∂xiin L1(Ω)−weak. (3.5)

On the other hand, since un is bounded in W 1LΦ (as it is weak∗ convergent),and since the embedding W 1LΦ → LΨ is compact, the set un is precompact inLΨ. Hence, by passing to a subsequence, if necessary, we can assume that un → vin LΨ, for some v ∈ LΨ. In particular, un → v in L1, and thus in D′(Ω). Hence,u = v in D′(Ω). This shows that u ∈ LΨ and un → u in LΨ. It follows that|un| → |u| in LΨ. Hence we may assume, without loss of generality, that un → ua.e. in Ω. Since ξ → Φ(|ξ|) is convex in ξ ∈ RN , we may use Theorem 2.1, Chapter8, [6], to obtain

f(u) =∫

ΩΦ(|∇u|)dx ≤ liminf

∫Ω

Φ(|∇un|)dx = lim inf f(un).

Thus f is weak∗ lower semicontinuous.

Lemma 3.3 The set Mµ is sequentially weak∗ closed.

Proof. We take a sequence un in Mµ with un ∗ u and prove that g(u) = µ.

To see this, we observe that

|g(un)− g(u)| ≤∫

Ω|Ψ(|un(x)|)−Ψ(|u(x)|)| dx,

and then

|g(un)− g(u)| ≤∫

Ω

∫ |u(x)|+|un(x)−u(x)|

|u(x)|ψ(t) dt dx

≤∫

Ωψ(|u(x)|+ |un(x)− u(x)|)|un(x)− u(x)| dx.

(3.6)

On the other hand, if v ∈ W 1LΦ, then, because of the embedding theoremv ∈ LΦ∗ . Let K be such that ‖v‖Φ∗ ≤ K and choose T > 0 such that

Ψ(t) ≤ Φ∗( t

2K

), t ≥ T

(recall that Ψ grows essentially more slowly than Φ∗ ). Let

ΩK =x : |v(x)| ≤ T

2

.

Then the above imply∫Ω

Ψ(2(|v(x)|))dx ≤∫

ΩKΨ(2(|v(x)|))dx +

∫Ω\ΩK

Φ∗

(|v(x)|K

)dx

≤ Ψ(T )|Ω|+∫

ΩΦ∗

(|v(x)|K

)dx

≤ Ψ(T )|Ω|+ 1;


hence 2v ∈ LΨ and∫Ω

Ψ∗(ψ(|v|)) dx =∫

Ω

∫ ψ(|v|)

0ψ−1(τ)dτdx

≤∫

Ω|v|ψ(|v|)dx ≤

∫Ω

Ψ(2|v|)dx.

and hence ψ(|v|) ∈ LΨ∗ . Inequality (3.6) therefore implies, via Holder’s inequalityin Orlicz spaces (see, [1]), that

|g(un)− g(u)| ≤ 2‖ψ(|u|+ |un − u|)‖Ψ∗‖un − u‖Ψ. (3.7)

It follows, since the sequence |un − u| is bounded in W 1LΦ, that the sequenceψn = ψ(|un(·)− u(·)|+ |u(·)|)

is uniformly bounded in LΨ∗ . For if ‖ψn‖Ψ∗ > 2,

then by the convexity of Ψ∗ we have that

2‖ψn‖Ψ∗

∫Ω

Ψ∗(ψn(x)) dx ≥∫

ΩΨ∗(

2ψn(x)‖ψn‖LΨ∗

)dx > 1

(recall again the definition of the Luxemburg norm) and hence,

12‖ψn‖Ψ∗ ≤

∫Ω

Ψ∗(ψn(x))dx

≤∫

Ωψ(|un(x)− u(x)|+ |u(x)|))(|un(x)− u(x)|+ |u(x)|))dx

≤∫

ΩΨ(2(|un(x)− u(x)|+ |u(x)|))dx

(compare the calculations following formula (3.6)). It also follows from the embed-ding theorem, Theorem 2.2, that

‖|un − u|+ |u|‖LΦ∗ ≤ K, ∀n,

where K is a constant.Using the fact that Ψ grows essentially more slowly than Φ∗ choose T > 0

such that (here we repeat earlier calculations)

Ψ(t) ≤ Φ∗

(t

2K

), t ≥ T.

Further let

Ωn =x : |un(x)− u(x)|+ |u(x)| ≤ T

2

.


Then the above imply∫Ω

Ψ(2(|un(x)− u(x)|+ |u(x)|))dx

≤∫

ΩnΨ(2(|un(x)− u(x)|+ |u(x)|))dx

+∫

Ω\ΩnΦ∗

(|un(x)− u(x)|+ |u(x)|

K

)dx

≤ Ψ(T )|Ω|+∫

ΩΦ∗

(|un(x)− u(x)|+ |u(x)|

K

)dx

≤ Ψ(T )|Ω|+ 1,

where |Ω| is the measure of Ω. Thus g(un)→ g(u) by formula (3.7). Hence g(u) =µ, and u ∈Mµ and Mµ is weak∗ closed.

Remark. The arguments just used also imply that g is strongly continuous andcompact, thus g is in fact completely continuous on W 1

0LΦ.

3.2 The minimization problem

The following theorem provides the existence of solutions of the minimizationproblem (3.3).

Theorem 3.1 The minimization problem (3.3) has a solution.

Proof. It follows from the properties of the functional g that Mµ 6= ∅. We chooseun ⊂Mµ such that

f(un)→ α.

The coercivity of f therefore implies that un is bounded in W 10LΦ which is in the

dual of a separable Banach space. Hence un has a weak∗ convergent subsequenceconverging to say u ∈W 1

0LΦ and u ∈Mµ since Mµ is weak∗ closed. By the weak∗

lower semicontinuity of f , f(u) = lim f(un) = α. u is therefore a solution of ourconstrained minimization problem.

As f is a proper functional (i.e. it assumes finite values) it follows, of course,that α < +∞.

3.3 Differentiability properties

Let us assume that the functionals f and g are as above. Let us show that f (witha condition on Φ∗) and g are C1 on W 1

0EΦ and W 10LΦ, respectively.

Lemma 3.4 Assume that Φ∗ satisfies a ∆2 condition. Then the functionalf : W 1

0EΦ → R is C1.


Proof. We observe that for u, h ∈W 10EΦ and t > 0

1t

(f(u+ th)− f(u)) =∫

Ω

1t

∫ |∇u+t∇h|

|∇u|sA(s2)dsdx.

On the other hand, as t→ 0,

|∇u+ t∇h| → |∇u|

in LΦ and hence in L1 (see [17]) and thus via a subsequence may be assumed toconverge almost everywhere. An easy calculation shows that∣∣∣∣∣1t

∫ |∇u+t∇h|

|∇u|sA(s2)ds

∣∣∣∣∣ ≤ φ(|∇u|+ |∇h|)|∇h|

with φ(|∇u| + |∇h|) ∈ LΦ∗ and |∇h| ∈ LΦ and hence φ(|∇u| + |∇h|)|∇h| ∈L1, because of a form of Holder’s inequality in Orlicz spaces ([1]). Thus, by thedominated convergence theorem,

limt→0

1t

(f(u+ th)− f(u)) =∫

Ω|∇u|A(|∇u|2)

∇u · ∇h|∇u| dx

=∫

ΩA(|∇u|2)∇u · ∇hdx

= f ′(u, h).

We note that this Gateaux derivative is linear with respect to h. We also have thatf ′ is continuous. Indeed if we let un ⊂ W 1

0EΦ be a sequence such that un → uand consider

|〈f ′(un)− f ′(u), h〉| =∣∣∣∣∫

Ω(A(|∇un|2)∇un −A(|∇u|2)∇u) · ∇hdx

∣∣∣∣ , (3.8)

then a lengthy exercise using Egoroff’s theorem shows that there exists a sequenceδn ⊂ R+ with δn → 0 such that

sup‖h‖1,Φ≤1

∣∣∣∣∫Ω

(A(|∇un|2)∇un −A(|∇u|2)∇u) · ∇hdx∣∣∣∣

≤ ‖φ(|∇un|)− φ(|∇u|)‖Φ∗ + δn.

Since‖φ(|∇un|)− φ(|∇u|)‖Φ∗ → 0

as follows from Theorem 12, page 83 of [22] (it is here that we have used for thefirst time that Φ∗ satisfies a ∆2 condition), we obtain that

||f ′(un)− f ′(u)||(W 10EΦ)∗ → 0

and thus the lemma is proved.


We remark here, that if Φ also satisfies a ∆2 condition, then the effectivedomain of f is the whole space W 1

0LΦ and f is C1 there.The verification that g is C1 on W 1

0LΦ is somewhat analogous and proceedsusing reasoning similar to the above (that Φ∗ satisfies a ∆2 condition is not needed)and some of the reasoning employed in the proof of Lemma 3.3. In fact, we havethe following result.

Lemma 3.5 The functional g : W 10LΦ → R is of class C1 and

〈g′(u), v〉 =∫

Ωψ(u)vdx.

Proof. An outline of the proof is presented here. To prove that g is of class C1 onW 1LΦ ⊃W 1

0LΦ, we proceed as before, i.e. we check that g is Gateaux differentiableon W 1LΦ and g′ is continuous from W 1LΦ to [W 1LΦ]∗. Let u, h ∈ W 1LΦ andt ∈ (0, 1). Then

1t[g(u+ th)− g(u)] =

∫Ω

1t[Ψ(u(x) + th(x))−Ψ(u(x))]dx,

and1t[Ψ(u(x) + th(x))−Ψ(u(x))]→ ψ(u(x))h(x), as t→ 0,

for a.e. x ∈ Ω. On the other hand, as in (3.6),∣∣∣∣1t [Ψ(u(x) + th(x))−Ψ(u(x))]∣∣∣∣ ≤ 1

t

∫ |u(x)|+t|h(x)|

|u(x)|ψ(τ)dτ

≤ ψ(|u(x)|+ |h(x)|)|h(x)|.

Because |u|+ |h| ∈W 1LΦ, the arguments after (3.6) show that ψ(|u|+ |h|) ∈ LΨ∗ .Since |h| ∈ LΨ, ψ(|u|+ |h|)|h| ∈ L1(Ω), the dominated convergence can be appliedto yield

limt→0

1t[g(u+ th)− g(u)] =

∫Ωψ(u)hdx := 〈g′(u), h〉.

Now, we show that the mapping W 1LΦ → (W 1LΦ)∗, u 7→ g′(u), is continuous.We first consider the case where (2.1) holds, for which Φ∗ is defined (as an

N function) and we assume Ψ grows essentially more slowly than Φ∗. Let un → uin W 1LΦ. For h ∈W 1LΦ, by Holder’s inequality in Orlicz spaces,

|〈g′(un)− g′(u), h〉| ≤∫

Ω|ψ(un)− ψ(u)||h|dx

≤ C‖ψ(un)− ψ(u)‖(Φ∗)∗‖h‖Φ∗≤ C‖ψ(un)− ψ(u)‖(Φ∗)∗‖h‖1,Φ.

((Φ∗)∗ is the complementary function of Φ∗). Since this holds for all h,

‖g′(un)− g′(u)‖(W 1LΦ)∗ ≤ C‖ψ(un)− ψ(u)‖(Φ∗)∗ . (3.9)


Now, from the arguments in the proof of Lemma 3.3, we know that if v ∈ LΦ∗ ,then ψ(|v|) ∈ LΨ∗ and thus ψ(v) ∈ LΨ∗ . On the other hand, since Ψ growsessentially more slowly than Φ∗, (Φ∗)∗ grows essentially more slowly than Ψ∗

(Lemma 13.1, [15]), and therefore

LΨ∗ ⊂ E(Φ∗)∗

(cf. [15] or [1]). It follows that ψ(v) ∈ E(Φ∗)∗ for all v ∈ LΦ∗ . In other words, themapping u 7→ ψ(u) maps LΦ∗ into E(Φ∗)∗ . Now, we can apply Theorem 17.3 of[15] (with r being any positive number) to conclude that the above mapping is, infact, continuous from LΦ∗ to E(Φ∗)∗ .

Since un → u in LΦ∗ (note the embedding W 1LΦ → LΦ∗), it follows thatψ(un) → ψ(u) in E(Φ∗)∗ . In other words, ‖ψ(un) − ψ(u)‖(Φ∗)∗ → 0. Hence, using(3.9), we obtain the continuity of g′.

The proof in the case∫ ∞

1

Φ−1(τ)

τN+1N

dτ < ∞ is carried out in a similar way,

employing some obvious modifications.

Remark. We here point out that in the nonreflexive case W 10EΦ is not necessarily

weak* closed in W 1LΦ.

3.4 The main theorems

We now consider the eigenvalue problem (E) in a general setting, without imposingconstraints on the growth of Φ.

In the general situation, the functional f is not necessarily differentiable (andeven may assume infinite values) on the whole space. Since f is convex (and lowersemicontinuous) on W 1

0LΦ, we can replace the derivative of f as considered inTheorem 3.3 by the subdifferential ∂f of f , and use a nonsmooth version of theLiusternik theorem. Another reason for considering the general case is that, unlikethe case when Φ satisfies a polynomial growth condition, the derivative of f doesnot, in general, act on W 1

0LΦ continuously. Hence, we cannot apply Liusternik’stheorem directly.

Before proving the result, we formulate (E) as an eigenvalue problem for aninclusion, or equivalently, for a variational inequality. Note that in the smoothcase, (E) is formulated (at least formally) as the equation:

f ′(u)− λg′(u) = 0.

Replacing f ′(u) by the subdifferential ∂f(u), this equation is extended as theinclusion

(E1) ∂f(u)− λg′(u) 3 0.

The equivalent variational inequality form of this inclusion (cf. [26]) is

(E2)f(v)− f(u)− λ〈g′(u), v − u〉 ≥ 0, ∀v ∈W 1

0LΦu ∈W 1

0LΦ.


Again, λ is called an eigenvalue of (E2) if this variational inequality has a solutionu 6= 0.

Theorem 3.2 Suppose that Φ and Ψ, are N-functions as given above, and assumeΨ grows essentially more slowly than Φ∗ if (2.1) holds. Then for every constantµ > 0, there exists λ > 0 such that (E2) (and equivalently (E1)) has a positive(negative) solution u such that g(u) = µ.

Proof. We consider the constrained minimization problem (3.3). As checked inTheorem 3.1, for each µ > 0, (3.3) has a positive (negative) solution u. To establishthe existence of the Lagrange multiplier λ, we need the nonsmooth version ofLiusternik’s theorem, proved in [16] and stated in Section 2.2. In this case, (3.3) isjust the minimization stated in Kubrusly’s theorem and (2.4) is the same as ourvariational inequality (E2).

It follows from the results in Section 3.1 (Lemma 3.5) that G = g is differen-tiable on W 1

0LΦ(Ω) and G′ = g′ is given by

〈g′(u), v〉 =∫

Ωψ(u)vdx, ∀v ∈W 1

0LΦ.

Let u be a solution of (3.3) (whose existence is proved in Theorem 3.1). We onlyneed to check the nondegeneracy condition (2.3). First, it is clear that u 6= 0; henceψ(u) 6= 0 in Ω (since ψ is strictly increasing and ψ(0) = 0). Choosing v = 0, itfollows that v ∈ D(f) (f(0) = 0) and

〈g′(u), v − u〉 = −∫

Ωψ(u)udx < 0,

(again, we use the strict monotonicity of ψ and the fact that ψ(u) and u have thesame sign). This verifies the second condition of (2.3).

To check the first condition, we note that since ψ(u) 6= 0 on Ω, we can choosea function v0 ∈ C1

0 (Ω) such that∫Ωψ(u)v0dx > 0.

Since∫

Ω ψ(u)u <∞, there exists t sufficiently large such that∫Ωψ(u)(tv0)dx >

∫Ωψ(u)udx.

Choosing v = tv0, we have 〈g′(u), v − u〉 > 0. Moreover, v ∈ C10 (Ω) ⊂ W 1

0EΦ ⊂D(f), and we have verified the second condition of (2.3). The existence of theLagrange multiplier λ now follows from Kubrusly’s theorem. Choosing v = 0 in(E2) , we have −f(u) + λ〈g′(u), u〉 ≥ 0. Hence,

λ ≥ f(u)(∫

Ωψ(u)udx

)−1

> 0.


Theorem 3.3 Suppose that Φ and Ψ, are N-functions as given above, and assumeΨ grows essentially more slowly than Φ∗ (if (2.1) holds). Further assume that Φand Φ∗ satisfy a ∆2 condition. Then for every constant µ > 0 there exists λ > 0such that (E) has a positive (negative) solution u such that g(u) = µ.

If it is the case that Φ = Ψ, and Φ satisfies a global ∆2 condition, the set ofsuch eigenvalues is bounded below.

Proof. The proof proceeds as follows. We set

Mµ = u ∈W 10LΦ : g(u) = µ, and let α = inf

u∈Mµ

f(u).

We note that for every µ ≥ 0 Mµ 6= ∅ (this will follow from the continuity of g).Since in this case the functionals are differentiable, we may apply what has

been established above and find that∫ΩA(|∇uµ|2)∇uµ · ∇vdx− λµ

∫ΩB(u2

µ)uµvdx = 0, ∀v ∈W 10EΦ, (3.10)

i.e. uµ is a weak solution of (E). Choosing v = uµ in (3.10), we obtain thatλµ > 0. Also since f(uµ) = f(|uµ|) and g(uµ) = g(|uµ|), we may assume that uµis one-signed in Ω.

Let us next consider the special case that Φ = Ψ, and let λµ be as above. Itfollows from (3.10) that

λµ =

∫ΩA(|∇uµ|2)|∇uµ|2dx∫

ΩA(u2

µ)u2µdx

. (3.11)

SinceΦ(t) ≤ tφ(t) ≤ Φ(2t), t ≥ 0,

we obtain that ∫Ω

Φ(|∇uµ|)dx∫Ω

Φ(2|uµ|)dx≤ λµ ≤

∫Ω

Φ(2|∇uµ|)dx∫Ω

Φ(|uµ|)dx(3.12)

and using the Poincare inequality (Lemma 2.1) in (3.12) we obtain

λµ ≥

∫Ω

Φ(|∇uµ|)dx∫Ω

Φ(2d|∇uµ|)dx. (3.13)

Hence if 2d ≤ 1 (which is the case for domains Ω with small diameter) we findthat

λµ ≥ 1.


Otherwise, if Φ satisfies a ∆2 condition (see (1.2)), then

λµ ≥1

α(2d).

Since the lower bounds so obtained are independent of µ, we find that

λ1 = infλµ : µ ∈ R\0 > 0. (3.14)

In general, however, λ1 ≥ 0.These calculations also show that problem (E) has no non-trivial solutions

for λ < λ1.As an example we use the N−function

Φ(s) = es2 − 1

whose complementary function Φ∗ satisfies a ∆2 condition (see [15]) but Φ doesnot. Our result implies the existence of principal eigenvalues (in the sense madeprecise above) for the problem

−div(e|∇u|2∇u) = λeu

2u in Ω

u = 0 on ∂Ω.

4 An existence result

In this section we will show existence of solutions for the problem (N) in theintroduction. Here we shall assume that

λ1 = infu∈W 1

0LΦ\0

∫Ω

Φ(|∇u|)dx∫Ω

Φ(|u|)dx

is positive (which is the case when Φ satisfies a ∆2 condition, as was verifiedabove).

We have the following theorem:

Theorem 4.1 Suppose that Φ∗ satisfies a ∆2 condition. Then problem (N) has asolution provided that p satisfies the condition (1.1), with a < λ1.

Proof. Consider the functional G : W 10LΦ → R ∪ ∞ defined by

G(u) :=∫

ΩΦ(|∇u|) dx−

∫ΩP (u)dx, (4.1)

where

P (u)(x) =∫ u(x)

0p(x, s)ds.


We have already shown that the functional∫

Ω Φ(|∇u|) dx is weak∗ lower semicon-tinuous. On the other hand, if un ∗ u then un is bounded and hence precom-pact in LΦ. The Niemitskii operator P maps LΦ continuously to LΦ∗ and thus foru ∈W 1

0LΦ, P (u) ∈ LΦ∗ . Hence each subsequence of un has a subsequence with∫ΩP (un)dx→

∫ΩP (u)dx.

We hence have that G is weak∗ lower semicontinuous.We next show that G is coercive also. In this regard we observe that for any

u ∈W 10LΦ,

G(u) ≥∫

ΩΦ(|∇u|) dx− a

∫Ω

Φ(|u|) dx−∫

Ωbudx.

Hence

G(u) ≥(

1− a

λ1

)∫Ω

Φ(|∇u|)dx− 2‖b‖Φ∗‖u‖1,Φ.

It follows from Lemma 3.13 of [12] that (here it is needed that Φ∗ satisfies a ∆2condition) ∫

ΩΦ(|∇u|) dx

‖u‖1,Φ→∞ as ‖u‖1,Φ →∞,

and hence that the functional G is coercive. Thus G assumes its global minimumat say u, i.e. u satisfies

G(u) = minv∈W 1

0LΦ

G(v).

On the other hand, again, G(u) < ∞, and since |∫

Ω P (u)dx| <∞, we must havethat

∫Ω Φ(|∇u|)dx <∞.

The second term of the functional G is C1 and therefore, by using againKubrusly’s theorem, we obtain that u must satisfy the variational inequality

∫Ω

Φ(|∇v|)dx−∫

ΩΦ(|∇u|)dx−

∫Ωp(x, u)(v − u)dx ≥ 0,∀v ∈W 1

0LΦ

u ∈W 10LΦ,

or since Φ∗ satisfies a ∆2 condition, 〈G′(u), h〉 = 0 for all h ∈W 10LΦ, i.e. u ∈W 1

0LΦsatisfies ∫

ΩA(|∇u|2)∇u · ∇hdx−

∫Ωp(x, u)hdx = 0,

implying that u is a weak solution of problem (N).


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Received January 20, 1998

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On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting