UNIT 4: NONLINEAR ELLIPTIC PARTIALDIFFERENTIAL EQUATIONS.

ELLIPTIC EQUATIONS WITH CRITICAL GROWTH INTHE GRADIENT.

JOSE CARMONA

Contents

1. Introduction 12. Elliptic equations with bounded dependence on the gradient 33. Elliptic equations with critical growth in the gradient. An

approximative scheme 43.1. A priori bounds 53.2. Gradients convergence 63.3. Passing to the limit 84. Singular elliptic equations with critical growth in the

gradient 114.1. Existence 114.2. Nonexistence 135. Comparison principles and Bifurcation results 145.1. Uniqueness - Comparison Principle 145.2. Compactness 155.3. Continua of solutions 165.4. Sub and supersolutions. The logistic model 20References 21

1. Introduction

This notes correspond to the third session of the Unit 4. “Nonlin-ear elliptic partial differential equations” of the Doc-Course “PartialDifferential Equations: Analysis, Numerics and Control, 2018”.

Along this course we will present some results concerned with ex-istence, nonexistence, uniqueness and bifurcation results for elliptic

Date: April, 20th 2018. Universidad de Granada.1

2 J. CARMONA

equations with critical growth in the gradient whose simplest model is

−∆u+ µ(x)g(u)|∇u|2 = f(x).

This is a particular case of a more general equation of the form

−div((ai(x, u,∇u))i) = F (x, u,∇u),

with (ai)i elliptic and F having quadratic dependence on the gradient.We will assume along this notes that the equations are posed in abounded domain Ω ⊂ RN and we will consider homogenous Dirichletboundary conditions.

Moreover, with G replaced with any function name, G(x, u,∇u) al-ways denote to G(x, u(x),∇u(x)) where G(x, s, ξ) is a Caratheodoryfunction, i.e. G : Ω × R × RN → R is continuous in (s, ξ) for almostevery x ∈ Ω and it is measurable with respect to x for all (s, ξ) ∈ RN+1.

It makes no effort to obtain sufficient conditions on G to assure thata critical point u0 of the functional I : W 1,2

0 (Ω)→ R given by

I(u) =1

2

∫Ω

|∇u|2 −∫

Ω

G(x, u), u ∈ W 1,20 (Ω),

satisfies that ∫Ω

∇u0∇v =

∫Ω

Gu(x, u0)v, ∀v ∈ W 1,20 (Ω).

Thus, critical points of I correspond to weak solutions to−∆u = Gu(x, u), in Ω,

u = 0, on ∂Ω.

When the functional is

J(u) =1

2

∫Ω

A(u)|∇u|2 −∫

Ω

G(x, u), u ∈ W 1,20 (Ω),

for some bounded and continuously differentiable real function A, thenformally critical points correspond to weak solutions to

−div(A(u)∇u) + A′(u)|∇u|2 = Gu(x, u), in Ω,

u = 0, on ∂Ω.

Observe that only bounded tests functions are allowed if A′ is alsobounded. Observe also that, at least formally, in the particular caseA(s) = sq for some q ∈ (0, 1) then A′(s) is singular at s = 0.

We will give a quick review of some results concerned with existenceof solution to elliptic boundary value problems with critical growth inthe gradient (including the singular case) by means of approximatingwith problems that admits a bounded solution.

NEPDE. CRITICAL GROWTH IN THE GRADIENT AND BIFURCATION 3

There exists a wide bibliography on the matter and a small part ofit has been used to prepare this notes, [2], [5], [6], [10], [13], [14], [17],[19], [20] and [21].

2. Elliptic equations with bounded dependence on thegradient

In this section we recall the result of [29] where general conditions onCaratheodory functions ai, i = 1, . . . , N , and F are obtained in orderto assure existence of weak solution to

(2.1)

−div((ai(x, u,∇u))i) = F (x, u,∇u), in Ω,

u = 0, on ∂Ω.

Theorem 2.1 (Leray-Lions). Assume that p ∈ (1,+∞) and

(1) there exists α > 0 such thatN∑i=1

ai(x, s, ξ)ξi ≥ α|ξ|p,

(2) there exists β > 0 such that |ai(x, s, ξ)| ≤ β(|s|p−1 + |ξ|p−1

),

for all i ∈ 1, . . . , N.

(3)N∑i=1

(ai(x, s, ξ)− ai(x, s, η)) (ξi − ηi) > 0 if ξ 6= η

(4) There exists f ∈ Lp′(Ω) such that |F (x, s, ξ)| ≤ f(x).

Then (2.1) admits a weak solution u ∈ W 1,p0 (Ω), i.e.∫

Ω

(ai(x, u,∇u))i∇v =

∫Ω

F (x, u,∇u)v, ∀v ∈ W 1,p0 (Ω)

Idea of the proof. Exercise. (1) Define A : W 1,p0 (Ω)→ W−1,p(Ω) by

A(v) = −div((ai(x, v,∇v))i)− F (x, v,∇v) ∈ W−1,p(Ω), i.e.

〈A(v), w〉 =

∫Ω

(ai(x, v,∇v))i∇w −∫

Ω

F (x, v,∇v)w, ∀w ∈ W 1,p0 (Ω).

(2) A is coercive, i.e. lim‖v‖

W1,p0 (Ω)

→+∞

〈A(v), v〉‖v‖W 1,p

0 (Ω)

= +∞.

(3) A is pseudomonotone, i.e., A maps bounded sets to boundedsets and whenever vn → v weakly in W 1,p(Ω) and satisfy thatlim supn〈A(vn), vn − v〉 ≤ 0 then

lim infn〈A(vn), vn − w〉 ≥ 〈A(v), v − w〉, ∀w ∈ W 1,p

0 (Ω).

(4) Surjectivity Theorem. Since W 1,p0 (Ω) is a Banach reflexive, sep-

arable space and A is a pseudomonotone coercive operator thenA is surjective.

4 J. CARMONA

3. Elliptic equations with critical growth in thegradient. An approximative scheme

We will deal along this notes with a simple model example of anelliptic equation having lower order terms with quadratic growth withrespect to the gradient without variational structure. More preciselywe consider

(3.1)

−div(A(x)∇u) + g(u)µ(x)|∇u|2 = f(x, u), in Ω,

u = 0, on ∂Ω,

where A(x) = (aij(x)) is a square matrix with aij ∈ L∞(Ω) such that

(A1) there exists α > 0 with A(x)ξ · ξ ≥ α|ξ|2.

We also consider µ(x) ∈ L∞(Ω), g : R → R continuous and f aCharatheodory function.

Definition 3.1. We say that u ∈ W 1,20 (Ω) is a weak solution to (3.1)

if f(x, u), g(u)µ(x)|∇u|2 ∈ L1(Ω) and∫Ω

A(x)∇u · ∇v +

∫Ω

g(u)µ(x)|∇u|2v =

∫Ω

f(x, u)v,

forall v ∈ W 1,20 (Ω) ∩ L∞(Ω).

In order to prove existence of solution to (3.1) a possible alternativeis to consider a sequence Fn, satisfying Leray-Lions conditions, suchthat Fn(x, s, ξ) approximates, as n → ∞, to f(x, s) − g(s)µ(x)|ξ|2.Thus we have a sequence un of solutions to approximative problemsand the question is: does un converge to a solution of (3.1)?

More precisely, it is assured (by means of Leray-Lions Theorem) theexistence of un ∈ W 1,2

0 (Ω) solution to(3.2)−div(A(x)∇u) +

g(u)µ(x)|∇u|2

1 + 1n|g(u)µ(x)||∇u|2

=f(x, u)

1 + 1n|f(x, u)|

, in Ω,

u = 0, on ∂Ω.

In order to answer the previous question it is important to provesome “a priori estimates”. Thus, if the sequence un is bounded inW 1,2

0 (Ω), up to a subsequence, it is weakly convergent to u ∈ W 1,20 (Ω).

This provide us with a candidate to be a weak solution to (3.1).Moreover, the weak convergence also allow to assure that∫

Ω

A(x)∇un · ∇v →∫

Ω

A(x)∇u · ∇v, ∀v ∈ W 1,20 (Ω).

NEPDE. CRITICAL GROWTH IN THE GRADIENT AND BIFURCATION 5

Therefore, in order to pass to the limit we only have to analyze thelower order term

Fn(x, un,∇un) =f(x, un)

1 + 1n|f(x, un)|

− g(un)µ(x)|∇un|2

1 + 1n|g(un)µ(x)||∇un|2

.

A first step is to determine if Fn(x, un,∇un) is a.e. convergent whichleads, since un(x) → u(x) a.e. x ∈ Ω to the following question: does∇un a.e. converge to ∇u?

Those problems will be analyzed in the following subsections.

3.1. A priori bounds. In this section we assume that N ≥ 3 and fsatisfies

(3.3) |f(x, s)| ≤ f1(x) ∈ L2NN+2 (Ω).

In addition we assume that g and µ satisfy

(3.4) (g(s)sµ(x))− ≥ −α1, 0 ≤ α1 < α.

Lemma 3.2. Assume (A1), (3.3), (3.4) and that un ∈ L∞(Ω) then unis bounded in W 1,2

0 (Ω).

Proof. Since un ∈ L∞(Ω) we can take un as test function in (3.2) andusing (A1) we obtain that

α‖un‖2W 1,2

0 (Ω)≤∫

Ω

|f(x, un)||un| −∫

Ω

(g(un)unµ(x))−|∇un|2

In particular, using (3.3), (3.4) and Holder inequality

(α− α1)‖un‖2W 1,2

0 (Ω)≤ ‖f1‖

L2NN+2 (Ω)

‖un‖L2∗ (Ω).

Finally, using Sobolev inequality we deduce that

‖un‖W 1,20 (Ω) ≤

1

S(α− α1)‖f1‖

L2NN+2 (Ω)

,

with S = infu∈W 1,2

0 (Ω)\0

‖u‖W 1,20 (Ω)

‖u‖L2∗ (Ω)

.

Exercise 1. Prove Lemma 3.2 with condition (3.3) replaced with

(3.5) |f(x, s)| ≤ f1(x) + f2(x)|s|p,

with p < 2∗ − 1, f1 ∈ L2NN+2 (Ω) and f2 ∈ L

2∗2∗−p−1 (Ω).

Exercise 2. In view of the nonlinear nature of the problem one mayinfer that nonlinear test functions may give better results. Improveconditions (3.3) and (3.4) using as test function a convenient nonlinearfunction of un (see [21] and [15])

6 J. CARMONA

Remark 3.3. Observe that actually under hypotheses of Lemma 3.2it is proved the following a priori bound for bounded solutions of (3.1):

‖u‖W 1,20 (Ω) ≤

1

S(α− α1)‖f1‖

L2NN+2 (Ω)

.

In order to apply Lemma 3.2 we need to prove that un ∈ L∞(Ω).This is consequence of the following Stampacchia Lemma [34].

Lemma 3.4. Assume that u ∈ W 1,20 (Ω) is a solution to −div(A(x)∇u) =

f1(x) with f1 ∈ Lp(Ω) for some p > N2

. Then u ∈ L∞(Ω).

Exercise 3. Prove Lemma 3.4.

We observe that, since −div(A(x)∇un) = Fn and |Fn| ≤ 2n then,from Lemma 3.4 we deduce that un ∈ L∞(Ω) and from Lemma 3.2 thatun is bounded in W 1,2

0 (Ω). Thus there exists u ∈ W 1,20 (Ω) such that,

up to a subsequence

• un weakly converges to u in W 1,20 (Ω).

• un strongly converges to u in Lp(Ω) for every 1 ≤ p < 2∗.• un(x)→ u(x) for a.e. x ∈ Ω.

The next step is to give sufficient conditions on f, g, µ to prove thea.e. convergence of the gradients of un in order to pass to the limitobtaining the equation satisfied by u.

Remark 3.5. A similar proof of the previous lemma allows to assurethat if −∆un = fn with the sequence fn is bounded in Lp(Ω) for somep > N/2 then ‖un‖L∞(Ω) ≤ c∞ for some positive constant c∞. Inaddition, for ∂Ω smooth enough, applying [27, Theorem 6.1] one candeduce that the sequence un is bounded in C0,α(Ω). Consequently,Ascolı-Arzela Theorem assures that un possesses a subsequence con-verging in C(Ω).

3.2. Gradients convergence. In this subsection we prove a resultfrom [19] concerning with the convergence of the sequence of gradientsof un.

Lemma 3.6. Assume that un is a bounded sequence in W 1,20 (Ω) sat-

isfying −div(A(x)∇un) = fn with fn bounded in L1(Ω). Then ∇unstrongly converges in (Lq(Ω))N for every 1 ≤ q < 2.

Proof. First of all let us denote u ∈ W 1,20 (Ω) such that, up to a sub-

sequence, un weakly converges to u in W 1,20 (Ω). For every δ > 0 and

NEPDE. CRITICAL GROWTH IN THE GRADIENT AND BIFURCATION 7

1 ≤ q < 2 we observe that∫Ω

|∇(un − u)|q =

∫|un−u|≥δ

|∇(un − u)|q +

∫|un−u|<δ

|∇(un − u)|q

≤||un − u| ≥ δ|1−q2

(∫Ω

|∇(un − u)|2) q

2

(3.6)

+ |Ω|1−q2

(∫|un−u|<δ

|∇(un − u)|2) q

2

.

Since un → u in L1(Ω) then ||un − u| ≥ δ| → 0 as n → ∞, andthe first term of (3.6) vanishes as n→∞.

In particular, in order to prove that∇un strongly converges in (Lq(Ω))N

for every 1 ≤ q < 2 we only have to show that

(3.7) limδ→0

(lim supn→∞

∫|un−u|<δ

|∇(un − u)|2)

= 0.

Observe that

α

∫|un−u|<δ

|∇(un − u)|2 ≤∫|un−u|<δ

A(x)∇(un − u)∇(un − u)

=

∫Ω

A(x)∇un∇Tδ(un − u)−∫|un−u|<δ

A(x)∇u∇(un − u)

≤∫

Ω

fn(x)Tδ(un − u) +

∫Ω

|A(x)∇u∇(un − u)| .

Here we have used Tδ(un − u) as test in the equation satisfied by un.Thus, using the weak convergence in W 1,2

0 (Ω) we deduce that

lim supn→∞

∫|un−u|<δ

|∇(un − u)|2 ≤ δ

αlim supn→∞

‖fn‖L1(Ω)

Taking limits as δ → 0 we finally obtain, using that fn is boundedin L1(Ω) and un is bounded in W 1,2

0 (Ω), we obtain (3.7).

Remark 3.7. Observe that dealing with problem (3.2) fn is givenby Fn(x, un,∇un). Thus, some particular cases where we can use theprevious Lemma are:

(1) Assume that g(s)µ(x) ≡ 0 (semilinear case). Then condition(3.5) is enough to have that∇un strongly converges in (Lq(Ω))N

for every 1 ≤ q < 2. In this case, it is well known that actuallyun strongly converges in W 1,2

0 (Ω).(2) In the general case we have to add, in addition to (3.5), a con-

dition on g. For example, if g(un) is bounded in L∞(Ω) then

8 J. CARMONA

Fn is bounded in L1(Ω). This is the case if un is bounded inL∞(Ω).

(3) In the case g, µ, un ≥ 0, taking Tδ(un) as test function, andusing Fatou lemma as δ → 0 it is possible to prove that Fn isbounded in L1(Ω).

(4) Using nonlinear test functions, sometimes it is possible to provethat actually un strongly converges in W 1,2

0 (Ω).

3.3. Passing to the limit. In order to pass to the limit in (3.2) weobserve that the weak convergence in W 1,2

0 (Ω) implies that

∫Ω

A(x)∇un∇ϕ→∫

Ω

A(x)∇un∇ϕ,

for every ϕ ∈ W 1,20 (Ω). Moreover, using the Sobolev embedding and

(3.5) we also have

∫Ω

f(x, un)

1 + 1n|f(x, un)|

ϕ→∫

Ω

f(x, u)ϕ.

Thus, in order to prove that u solves (3.1) it remains to prove that

(3.8)

∫Ω

g(un)µ(x)|∇un|2

1 + 1n|g(un)µ(x)|∇un|2|

ϕ→∫

Ω

g(u)µ(x)|∇u|2ϕ,

for every ϕ ∈ W 1,20 (Ω) ∩ L∞(Ω). Since we know the a.e. convergence

we can obtain (3.8) if we can use Lebesgue Theorem. For example, ifun is bounded in L∞(Ω) and un strongly converges in W 1,2

0 (Ω) then,up to a subsequence, |∇un|2 is dominated in L1(Ω) and we can pass tothe limit using Lebesgue theorem.

Here we present an alternative way to pass to the limit using Fatoulemma, see [16], when un is bounded in L∞(Ω).

Theorem 3.8. Assume that the sequence un of solutions to (3.2) isbounded in L∞(Ω). Then the weak limit satisfies (3.1).

NEPDE. CRITICAL GROWTH IN THE GRADIENT AND BIFURCATION 9

Proof. Indeed, in this case, given ϕ ≥ 0 we can take exp(H(un) −H(u))ϕ as test function (H ∈ C1 nondecreasing ) and we obtain that∫

Ω

A(x)∇un (H ′(un)∇un −H ′(u)∇u) exp(H(un)−H(u))ϕ+∫Ω

A(x)∇un∇ϕ exp(H(un)−H(u))+∫Ω

g(un)µ(x)|∇un|2

1 + 1n|g(un)µ(x)|∇un|2|

exp(H(un)−H(u))ϕ

=

∫Ω

f(x, un)

1 + 1n|f(x, un)|

exp(H(un)−H(u))ϕ

In particular∫Ω

(H ′(un)A(x)∇un∇un +

g(un)µ(x)|∇un|2

1 + 1n|g(un)µ(x)|∇un|2|

)eH(un)−H(u)ϕ

= −∫

Ω

A(x)∇un∇ϕ exp(H(un)−H(u))+

+

∫Ω

H ′(u)A(x)∇un∇u exp(H(un)−H(u))ϕ

+

∫Ω

f(x, un)

1 + 1n|f(x, un)|

exp(H(un)−H(u))ϕ

→ −∫

Ω

A(x)∇u∇ϕ+

∫Ω

H ′(u)A(x)∇u∇uϕ+

∫Ω

f(x, u)ϕ

In order to use Fatou Lemma we observe that

H ′(un)A(x)∇un∇un+g(un)µ(x)|∇un|2

1 + 1n|g(un)µ(x)|∇un|2|

≥ α(H ′(un)− C|g(un)|)|∇un|2

Thus, taking H ′(s) ≥ Cα|g(s)| we can use Fatou lemma and we obtain

that ∫Ω

H ′(u)ϕA(x)∇u∇u+ g(u)µ(x)|∇u|2ϕ

≤ lim inf

∫Ω

(H ′(un)A(x)∇un∇un +

g(un)µ(x)|∇un|2

1 + 1n|g(un)µ(x)|∇un|2|

)eH(un)−H(u)ϕ

= −∫

Ω

A(x)∇u∇ϕ+

∫Ω

H ′(u)A(x)∇u∇uϕ+

∫Ω

f(x, u)ϕ

10 J. CARMONA

That is,∫Ω

A(x)∇u∇ϕ+

∫Ω

g(u)µ(x)|∇u|2ϕ ≤∫

Ω

f(x, u)ϕ

Similarly, taking exp(H(u)−H(un))ϕ as test function∫Ω

A(x)∇un (H ′(u)∇u−H ′(un)∇un) exp(H(u)−H(un))ϕ+∫Ω

A(x)∇un∇ϕ exp(H(u)−H(un))+∫Ω

g(un)µ(x)|∇un|2

1 + 1n|g(un)µ(x)|∇un|2|

exp(H(u)−H(un))ϕ

=

∫Ω

f(x, un)

1 + 1n|f(x, un)|

exp(H(u)−H(un))ϕ

In particular∫Ω

(H ′(un)A(x)∇un∇un −

g(un)µ(x)|∇un|2

1 + 1n|g(un)µ(x)|∇un|2|

)eH(u)−H(un)ϕ

=

∫Ω

A(x)∇un∇ϕ exp(H(u)−H(un))+∫Ω

H ′(u)A(x)∇un∇u exp(H(u)−H(un))ϕ

−∫

Ω

f(x, un)

1 + 1n|f(x, un)|

exp(H(u)−H(un))ϕ

→∫

Ω

A(x)∇u∇ϕ+

∫Ω

H ′(u)A(x)∇u∇uϕ−∫

Ω

f(x, u)ϕ

Now we observe that

H ′(un)A(x)∇un∇un−g(un)µ(x)|∇un|2

1 + 1n|g(un)µ(x)|∇un|2|

≥ α(H ′(un)− C|g(un)|)|∇un|2

Thus, since H ′(s) ≥ Cα|g(s)| we can use Fatou lemma and we obtain

that ∫Ω

H ′(u)ϕA(x)∇u∇u− g(u)µ(x)|∇u|2ϕ

≤ lim inf

∫Ω

(H ′(un)A(x)∇un∇un −

g(un)µ(x)|∇un|2

1 + 1n|g(un)µ(x)|∇un|2|

)eH(u)−H(un)ϕ

=

∫Ω

A(x)∇u∇ϕ+

∫Ω

H ′(u)A(x)∇u∇uϕ−∫

Ω

f(x, u)ϕ

NEPDE. CRITICAL GROWTH IN THE GRADIENT AND BIFURCATION 11

That is,∫Ω

A(x)∇u∇ϕ+

∫Ω

g(u)µ(x)|∇u|2ϕ ≥∫

Ω

f(x, u)ϕ,

and we finish the proof.

4. Singular elliptic equations with critical growth inthe gradient

In this section we consider the case where the function g is singularat zero. Let us illustrate the main results in the particular case g(s) =1/sγ. More precisely we consider the problem:

(4.1)

−div(A(x)∇u) + µ(x) |∇u|

2

uγ= f(x, u), in Ω,

u > 0, in Ω,

u = 0, on ∂Ω.

Singular problems also were studied in [1], [3], [9], [14] and [26].

4.1. Existence. Assume that f(x, s) ≡ f(x) i.e. consider the problem

(4.2)

−div(A(x)∇u) + µ(x) |∇u|

2

uγ= f(x), in Ω,

u > 0, in Ω,

u = 0, on ∂Ω.

The following existence result was obtained in [2].

Theorem 4.1. Let f in L2NN+2 (Ω) be such that

(4.3) mω(f)def= ess inf f(x) : x ∈ ω > 0, ∀ω ⊂⊂ Ω.

and suppose that γ < 2. Then there exists a solution u ∈ W 10 (Ω) to

problem (4.2).

In fact, condition (4.3) is only needed in the case γ > 1 (there is analternative proof in [14]).

Proof. The proof relies on an approximate scheme. Specifically, wedefine

gn(s)def=

1

sγs ≥ 1

n,

nγ+1s 0 < s ≤ 1

n,

0 s ≤ 0.

12 J. CARMONA

and fn = Tn(f). For every fixed n we can use Leray-Lions Theoremthat assures the existence of un ∈ W 1,2

0 (Ω) ∩ L∞(Ω) solution to

(4.4)

−div (A(x)∇un) + gn(un)µ(x)|∇un|2

1 + 1n|∇un|2

= fn in Ω,

un = 0 on ∂Ω,

Moreover un > 0 in Ω and the sequence is bounded in W 1,20 (Ω). But the

main property of the sequence un is that it is uniformly bounded frombelow, away from zero, in every compact set in Ω. This is consequenceof the following proposition proved in [2].

Proposition 4.2. Suppose that f ∈ L∞loc(Ω) satisfies (4.3). Let ω bea compactly contained open subset of Ω. Then there exists a constantcω > 0 such that every supersolution 0 < z ∈ W 1,2

loc (Ω) ∩ C(Ω) of theequation

(4.5) − div (A(x)∇z) + k|∇z|2

zγ= f in Ω,

satisfiesz ≥ cω in ω.

This proposition is proved performing the change of variable v =

ψ(z) with ψ(s) =∫ 1

se−

H(t)α dt and H(s) = k

∫ s1t−γdt+ log sα.

One can prove that v ∈ W 1loc(Ω) and it is a subsolution to

−div (A(x)∇v) + f(x) b(v) = 0 in Ω.

with b(s) = e−H(ψ−1(s))

α − 1. Observe that b(s)s

is nondecreasing for larges > 0. Moreover, since γ < 2 then the function b(s) satisfies the well-known Keller-Osserman condition i.e., there exists t0 > 0 such that

(4.6)

∫ +∞

t0

dt√2∫ t

0b(s)ds

< +∞.

By applying [28, Theorem 7] we derive that for every ω ⊂⊂ Ω, thereexists Cω > 0 such that

v ≤ Cω in ω.

Therefore, undoing the change

z ≥ ψ−1(Cω) = cω > 0 in ω ,

as desired.Now it is possible to prove that, for every k > 0, Tk(un) → Tk(u)

strongly in W 1,2loc (Ω), where u denotes the weak limit of un. Reason-

ing as in [18], we consider the function ϕλ(s) = seλs2

and we choose

NEPDE. CRITICAL GROWTH IN THE GRADIENT AND BIFURCATION 13

ϕλ(Tk(un) − Tk(u))φ, with φ ∈ C∞c (Ω), φ ≥ 0 and λ large enough, astest function in the equation satisfied by un.

Then, using that∫Ω

|∇Gk(un)|2 ≤ S2

α2

(∫un≥k

f2NN+2

)1+ 2N

,

we can prove that un is strongly convergent in W 1loc(Ω).

Finally, using Vitali theorem we can pass to the limit.

Exercise 4. Consider f ∈ Lp(Ω) for some p > N2

. Give an alternativeproof of the previous result, passing to the limit using Fatou Lemma.Observe that taking 0 ≤ ϕ ∈ H1

0 (Ω)∩Cc(Ω) as test function and usingFatou Lemma, the function u (weak limit of un) satisfies∫

Ω

A(x)∇u∇φ+

∫Ω

µ(x)|∇u|2

uγφ ≤

∫Ω

f(x)φ.

In order to obtain the reverse inequality define the function

H(t) =

∫ t

1

‖µ‖L∞(Ω)

sγds, t > 0,

and use that un is also bounded in L∞(Ω) and Proposition 4.2 withω = supp φ to prove that you can take eH(u)−H(un)φ as test function.

4.2. Nonexistence. For γ ≥ 2 there is no energy solution as is de-duced from the following result taken from [35].

Lemma 4.3. Assume that u ∈ W 1,20 (Ω) \ 0. Then |∇u|2

|u|γ 6∈ L1(Ω) for

every γ ≥ 2. In particular, problem (4.1) does not admit solution.

Proof. Observe that

∫Ω

|∇u|2

|u|γ≥∫

Ω

|∇u|2

(|u|+ ε)γ=

∫Ω

∣∣∣∣ ∇u(|u|+ ε)

γ2

∣∣∣∣2=

∫Ω

∣∣∣∣∇(∫ u

0

ds

(|s|+ ε)γ2

)∣∣∣∣2 ≥ C

∫Ω

∣∣∣∣∫ u

0

ds

(|s|+ ε)γ2

∣∣∣∣2≥ C

∫|u|>δ

(∫ |u|0

ds

(|s|+ ε)γ2

)2

≥ C

∫|u|>δ

∣∣∣∣∣∫ |u|

0

ds

(s+ ε)γ2

∣∣∣∣∣2

≥ C

∫|u|>δ

∣∣∣∣∫ δ+ε

ε

dt

tγ2

∣∣∣∣2 ≥ C||u| ≥ δ|∣∣∣∣∫ δ

ε

dt

tγ2

∣∣∣∣2

14 J. CARMONA

Using that u 6= 0 we can choose δ > 0 such that ||u| ≥ δ| > 0.

Moreover, since γ > 2 we have that

∫ δ

ε

dt

tγ2

diverges as ε→ 0.

Exercise 5. Prove a similar result with 1uγ

replaced with a positive

function g : (0,+∞)→ (0,+∞), decreasing near zero such that√g(s)

is not integrable at zero

An alternative proof of nonexistence for (4.1) is given in [2].

5. Comparison principles and Bifurcation results

In the last section we show a comparison principle for problems withquadratic lower order terms. As a consequence we show how to usebifurcation techniques or the sub and supersolution method to studysingular problems as (4.1) with a nonlinear right hand side f(λ, x, u)depending on a parameter λ. We will assume in this section that ∂Ωis smooth.

5.1. Uniqueness - Comparison Principle. A comparison principlefor general differential operators of the form

−div((ai(x, u,∇u))i) +H(x, u,∇u)

was stablished in [12] and slightly improved in [11]. In both cases, when(ai(x, s, ξ))i = ξ, conditions imposed to H require that ∂sH ≥ 0.

In [10] was proved a comparison principle for (ai(x, s, ξ))i = a(s)ξand H(x, u,∇u) = g(u)|∇u|2 for some nonnegative continuous functiong in (0,+∞). In this case, the authors imposed the integrability ofg(s)a(s)

at zero. This result handles the case that g is singular at zero

(which necessarily implies that ∂sH 6≥ 0). However, their techniquesrequire strongly that the function H and the differential operator donot depend on x.

Some further extension, dealing with uniqueness, was done in [7] inthe case (ai(x, s, ξ)) = ξ and H(x, u,∇u) = −d(x)u−µ(x)|∇u|2−h(x)for some d, h ∈ Lp(Ω), p > N/2, d ≤ 0 and µ ∈ L∞(Ω) (see [8]for an slightly improvement with more general ai and H). It is onceagain imposed that ∂sH ≥ 0. Moreover, in some particular cases, with∂sH < 0 (d(x) > 0) they prove a multiplicity result (see Theorem 1.3in [7]), that is, no uniqueness result is expected imposing only that∂sH < 0.

More recently, in [4] it is proved a comparison principle in the case(ai(x, s, ξ)) = ξ for a particular class of functions H(x, u,∇u) whichare continuous at u = 0 and that may be decreasing on u.

NEPDE. CRITICAL GROWTH IN THE GRADIENT AND BIFURCATION 15

We present here a simplified version of the main result in [6] wherewe prove the comparison principle for (ai(x, s, ξ)) = a(s)A(x)ξ andH dependending on x and being singular at u = 0 on the gradientquadratic part. More precisely we consider here

(5.1)

−div(a(u)∇u) + g(u)µ(x)|∇u|2 = f(x) in Ω

u = 0 on ∂Ω.

We say that u ∈ W 1,20 (Ω) with u > 0 is a subsolution (respectively,

a supersolution) of (5.1) if a(u)A(x)∇u ∈ L2(Ω)N , g(u)|∇u|2 ∈ L1(Ω)and ∫

Ω

a(u)A(x)∇u∇φ+

∫Ω

g(u)µ(x)|∇u|2φ ≤∫

Ω

fφ ,

for every φ ∈ W 1,20 (Ω) ∩ L∞(Ω), (respectively, if the reverse inequality

holds). We prove the following theorem.

Theorem 5.1. Assume that for every ν > 0 there exist θ ≥ 0 and a

nonnegative function h ∈ C1((0,+∞)) with a(s)e−∫ s1h(t)a(t)

dt ∈ L1(0, 1)such that for a.e. x ∈ Ω and for every 0 < s < ν,

θ[a(s)∂s(g(s)µ(x)− h(s)) + (h(s)− 2a′(s))(g(s)µ(x)− h(s))](5.2)

−(g(s)µ(x)− h(s))2 ≥ 0.

If 0 < v1, v2 ∈ H10 (Ω)∩C(Ω) are respectively a sub and a supersolution

for (5.1) then v1 ≤ v2. As a consequence, we have uniqueness of C(Ω)solutions of (5.1).

5.2. Compactness. In order to study, from the point of view of bi-furcation theory, the problem

(5.3)

−∆u+ g(u)µ(x)|∇u|2 = f(λ, x, u) in Ω

u = 0 on ∂Ω.

we need some compactness properties for the operator T : C(Ω) →C(Ω) defined, for every w ∈ C(Ω), as the unique solution u ∈ W 1,2

0 (Ω)∩L∞(Ω) to the problem−∆u+ g(u)µ(x)|∇u|2 = f(λ, x, w) in Ω

u = 0 on ∂Ω.

T is well defined thanks to the comparison principle proved previously.The following lemma ensures the compactness properties required later.

16 J. CARMONA

Lemma 5.2. Assume that f(λn, x, wn) is bounded in Lq(Ω) with q > N2

and µ ∈ L∞(Ω). Let assume that 0 < un ∈ W 1,20 (Ω) ∩ C(Ω) satisfies

(5.4)

−∆un + µ(x)gn(un)|∇un|2 = f(λn, x, wn) in Ω,un = 0 on ∂Ω,

with 0 ≤ λn bounded in R, 0 ≤ wn bounded in C(Ω) and 0 ≤ gn asequence of functions in C((0,+∞)). Then, up to a subsequence, un isstrongly convergent in C(Ω) to u ∈ W 1,2

0 (Ω) ∩ C(Ω). If, in addition,λn → λ, wn → w in C(Ω), gn(s)→ g(s) uniformly in C([a, b]) for every

0 < a < b <∞, gn(s) ≤ h(s) for some h such that√h is integrable at

zero and f ≥ f0 with f0 satisfying (4.3), then u is a solution of problem

(5.5)

−∆u+ µ(x)g(u)|∇u|2 = f(λ, x, w) in Ω,u = 0 on ∂Ω .

Moreover, if the problem (5.5) admits a unique solution then the wholesequence un converges strongly to u in C(Ω).

5.3. Continua of solutions. In this section we study, from the pointof view of bifurcation theory, the problem

(5.6)

−∆u+ g(u)|∇u|2 = λup + f0(x) in Ω

u = 0 on ∂Ω.

Here we present the main results from [5]. In the case p = 1 we givesufficient conditions on g to have, as in the semilinear case, existence ofsolution for a bounded interval of values of the parameter λ. Roughlyspeaking, those conditions are related to how far from zero we can takeg in order to remain true the “semilinear type” result. On the otherhand, we also prove a regularizing effect, that is, existence of solutionin an unbounded interval of values for the parameter λ provided that gis sufficiently far from zero. We will assume that either g is continuousat zero or g(s) = 1

sγwith γ < 1.

Theorem 5.3. Assume p = 1, 0 f0 ∈ L2NN+2 (Ω).

(1) (No regularizing effect) If there are s0, δ0 > 0 such that

(5.7)s∫ s

0e∫ sr g(t)dtdr

≥ δ0, ∀s > s0,

then there exist λ∗, λ∗ > 0 such that (5.6) has no positive so-lution for λ > λ∗ and admits a positive solution for everyλ ∈ [0, λ∗).

NEPDE. CRITICAL GROWTH IN THE GRADIENT AND BIFURCATION 17

(2) (Regularizing effect) If there exist s1, c > 0, and γ < 1 such that

(5.8) g(s) ≥ c

sγ, ∀s ≥ s1,

then (5.6) admits a positive solution for every λ ∈ [0,+∞).

Observe that there is a gap between both conditions (5.7) and (5.8)

since, for instance, the function g(s) =c

s+ 1with c ≥ 1 satisfies neither

(5.7) nor (5.8).In the case 0 ≤ p < 1 we show that the behavior of g at infinity has

no influence in the solution set since there exists a positive solution forevery λ ∈ (0,+∞).

Theorem 5.4. If 0 ≤ p < 1, 0 ≤ f0 ∈ L2NN+2 (Ω)then (5.6) admits a

positive solution for every λ ∈ (0,+∞).

With respect to the case p > 1, we have the following result.

Theorem 5.5. Consider p > 1, 0 f0 ∈ L2NN+2 (Ω) and assume that

hypothesis (G) is satisfied.

i) If condition (5.8) holds with 0 ≤ γ < 2− p, then problem (5.6)admits a positive solution for every λ ∈ [0,+∞).

ii) If there are s0, δ0 > 0 such that

(5.9)sp∫ s

0e∫ sr g(t)dtdr

≥ δ0, ∀s > s0,

then there exist λ∗, λ∗ > 0 such that (5.6) admits a positivesolution for every λ ∈ [0, λ∗) and admits no positive solutionfor λ > λ∗.

If g is continuous in zero, we can handle the case f0 ≡ 0 and toobtain the following result.

Theorem 5.6. Assume f0 ≡ 0 and suppose that g ≥ 0 is continuousin the interval [0,+∞).

(1) (No regularizing effect) If p = 1 and (5.7) holds, then there existλ∗, λ∗ > µ1 such that (5.6) has no positive solution for λ > λ∗

and admits a positive solution for every λ ∈ (µ1, λ∗)(2) (Regularizing effect) If p = 1 and (5.8) holds, then (5.6) admits

a positive solution if and only if λ > µ1.(3) If 0 ≤ p < 1, then (5.6) admits a positive solution for every

λ ∈ (0,+∞).

18 J. CARMONA

(4) If p > 1 and there is a continuous non positive function h ∈L1(0,+∞) such that

(5.10) g(s) ≥ h(s) +p

s, ∀s ≥ 1,

then there exists λ∗ > 0 such that (5.6) has no solution forλ < λ∗.

We remark that cases (1)-(3) of the above result are new, while (4)is a slight improvement of [31] where it is required g(s) > q/s for larges, with q > p.

In [25] we also consider the following boundary value problem

(Pλ)

−∆u+ µ(x)g(u) |∇u|2 = λup + f0(x) in Ω,u = 0 on ∂Ω,

with µ ∈ L∞(Ω).Results concerning (Pλ) for λ 6= 0 were obtained in [5, 23] in the case

g(s) = 1/sγ where the model problem is

(Rλ)

−∆u+ µ(x)

|∇u|2

uγ= λup + f0(x) in Ω,

u = 0 on ∂Ω,

with µ(x) as a constant function. More precisely, with γ < 1 andγ+p < 2 the existence of a solution for each λ ≥ 0 was proved in [5] bymeans of topological methods and in [23] by using an approximativescheme.

However, the techniques employed in [5, 23] can not be applied inthe case µ(x) not constant or where p < 1 ≤ γ < 2. We deal with (Pλ)for a function g exhibiting a different behavior at zero and at infinity.

In particular, we are mainly interested in the case of functions g(s) =1/(sγ + sβ) with γ ≤ β. In this way, we consider the model problem

(Qλ)

−∆u+ µ(x)|∇u|2

uγ + uβ= λup + f0(x) in Ω,

u = 0 on ∂Ω,

as a natural extension of the problem (Rλ).We prove the following theorem in the case µ constant.

Theorem 5.7. Assume µ(x) = µ is constant and that f0 ∈ Lq(Ω) withq > N

2satisfies (4.3) . Then:

i) If 1 ≤ γ < 2 and 0 < p < 1 then problem (Qλ) admits at leastone solution for every λ ≥ 0.

NEPDE. CRITICAL GROWTH IN THE GRADIENT AND BIFURCATION 19

ii) If γ < 1 < β and 1 ≤ p, then there exists λ∗, λ∗ > 0 such that

(Qλ) admits no solution for λ > λ∗ and at least one solutionfor 0 ≤ λ < λ∗.

Moreover, there exists an unbounded continuum Σ of solutions of(Qλ), such that there exists uλ solution of (Qλ) with (λ, uλ) ∈ Σ forevery λ ≥ 0 (item i)) or every 0 ≤ λ < λ∗ (item ii)).

We would like to stress that in the case of item i), it is not required

assumptions on the parameter β. This is because in order to |∇u|2uγ+uβ

be an integrable function we only need the natural hypothesis γ < 2which is a condition at zero. In other words, the behavior of g atinfinity has not a role in the solutions set. Conversely, item ii) showsthat no regularizing effect take place since there is no solution for allpositive λ.

Moreover, observe that this theorem improve the results of [5, 23]since item i) with γ = β gives us existence results of the problem (Rλ)in the case γ > 1.

Furthermore, our techniques also allow us to work with non-constantfunction µ(x) when γ ≤ 1, γ + p < 2. In fact, if we suppose that thereexist positive constants m,M such that

(5.11) m ≤ µ(x) ≤M, a.e. x ∈ Ω,

we prove the following theorem.

Theorem 5.8. Assume that 0 < γ ≤ β ≤ 1, 0 < p < 2−β, f0 ∈ Lq(Ω)with q > N

2and (5.11) where M < 2 in the case γ = β = 1 and M > 0

otherwise. Then there exists an unbounded continuum Σ of solutionsof (Qλ), such that there exists uλ solution of (Qλ) with (λ, uλ) ∈ Σ forevery λ ≥ 0.

Note that this theorem with γ = β < 1 improves again the resultsof [5] since we can consider non-constant function µ(x). Furthermore,it improves also [23] except regularity of f0; in this work the authors

consider data f0 belonging to L2N

2N−γ(N−2) (Ω).In addition, since we deal with γ < β and the function g(s) = 1/(sγ+

sβ) behaves at infinity as 1/sβ do, we also show that the hypothesisp < 2 − β is a restriction in the behavior of g at infinity, rather thanin the singularity at zero.

We obtain the existence of the continuum in the above two theoremsby using a double approach. Initially, for a convenient sequence of ap-proximated problems, we can derive the existence of Σn by means of

20 J. CARMONA

Leray-Schauder degree techniques and Rabinowitz continuation theo-rem as in [5]. Secondly, we use a topological lemma to obtain a con-tinuum of solutions as the limit of this approximative scheme Σn. Itis also important to note that condition (4.3) becomes crucial whenapplying this approach in Theorem 5.7.

5.4. Sub and supersolutions. The logistic model. In [24] westudy existence, nonexistence and uniqueness of positive solutions tothe following nonlinear elliptic problem

(5.12)

−∆u+

|∇u|2

uγ= f(λ, u) in Ω,

u = 0 on ∂Ω,

with

(5.13) f(λ, s) = λsq or f(λ, s) = λs− sp, ∀s > 0.

The main results are the following:

Theorem 5.9. Assume that f(λ, u) = λu.

(1) If γ < 1, there exists a positive solution of (5.12) if and onlyif λ > λ1. Moreover, for λ > λ1 there exists a unique boundedpositive solution.

(2) If γ = 1 and k < 1, then there exists positive solution of (5.12)if and only if λ = λ1/(1 − k). In this case, there exist infinitepositive solutions.

(3) If γ = 1 and k ≥ 1, then (5.12) has no positive solution forλ > 0.

(4) If γ > 1 then (5.12) has no positive solution for λ > 0.

Theorem 5.10. Assume that f(λ, u) = λuq, 0 < q < 1.

(1) If γ < 1, there exists a unique bounded positive solution of(5.12) for every λ > 0.

(2) If γ = 1 and k ≤ q, there exists at most one positive solutionfor every λ > 0.

(3) If γ + q > 2, then (5.12) has no positive solution for λ > 0.

Theorem 5.11. Assume that f(λ, u) = λuq, 1 < q.

(1) If γ < 1 and γ + q < 2, there exists λ∗ > 0 such that (5.12)possesses a positive solution for every λ ≥ λ∗ and (5.12) doesnot possess any positive solution for every λ < λ∗.

(2) If γ ≥ 1 then (5.12) has no positive solution for λ > 0.

Theorem 5.12. Assume that f(λ, u) = λu− up, p > 1.

NEPDE. CRITICAL GROWTH IN THE GRADIENT AND BIFURCATION 21

(1) Any positive solution u of (5.12) is bounded and ‖u‖L∞(Ω) ≤λ1/(p−1).

(2) If γ < 1, there exists a positive solution of (5.12) if and only ifλ > λ1. In this case, the solution is unique.

(3) If γ = 1 and k < 1, if there exists positive solution of (5.12)then it is unique and λ > λ1/(1− k).

(4) If γ = 1 and k ≥ 1, then (5.12) has no positive solution forλ ≥ 0.

(5) If γ > 1, then (5.12) has no positive solution for λ ≥ 0.

To prove the existence results, we use the sub-supersolution methodfor weak-solution. We employ different arguments to show the non-existence results, some of them are an adequate extension of those in[2, 5, 31]. For γ ≥ 1 and γ + q > 2 we are able to prove that anypositive solution satisfies that |∇u|2/u2 ∈ L1(Ω), which is not possibleas was shown in [35].

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