Recently in the analysis of some new models, that are called electrorheological fluids, the following equation has been studied −∆p(x)u=f(x, u) in Ω

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

CONCENTRATION-COMPACTNESS PRINCIPLE FOR VARIABLE EXPONENT SPACES AND APPLICATIONS

JULI ´AN FERN ´ANDEZ BONDER, ANAL´IA SILVA

Abstract. In this article, we extend the well-known concentration - compact- ness principle by Lions to the variable exponent case. We also give some appli- cations to the existence problem for thep(x)-Laplacian with critical growth.

1. Introduction

When dealing with nonlinear elliptic equations with critical growth (in the sense of the Sobolev embeddings) the concentration - compactness principle by Lions, see [12], have been proved to be a fundamental tool for proving existence of solutions.

Just to cite a few references, we have [1, 2, 3, 7, 4, 11] but there is an impressive list of references on this topic.

Recently in the analysis of some new models, that are called electrorheological fluids, the following equation has been studied

−∆p(x)u=f(x, u) in Ω. (1.1)

The operator ∆_p(x)u := div(|∇u|^p(x)−2∇u) is called the p(x)-Laplacian. When p(x)≡pis the well-known p-Laplacian.

In recent years a vast amount of literature that deal with the existence problem for (1.1) with different boundary conditions (Dirichlet, Neumann, nonlinear, etc) have appeared. See, for instance [5, 6, 8, 13, 14] and references therein.

However, up to our knowledge, no results are available for (1.1) when the source term f is allowed to have critical growth at infinity (see the remark after the introduction for more on this). That is,

|f(x, t)| ≤C(1 +|t|^q(x))

with q(x)≤ p^∗(x) := N p(x)/(N−p(x)) (if p(x) < N) and {q(x) = p^∗(x)} 6=∅.

This article attempts to begin filling this gap. So, the objective is to extend the concentration - compactness principle by Lions to the variable exponent setting.

The method of the proof follows the lines of the ones in the original work of P.L. Lions and the main novelty in our result is the fact that we do not require the exponentq(x) to be critical everywhere. Moreover, we show that the delta masses are concentrated in the set whereq(x) is critical.

2000Mathematics Subject Classification. 35J20, 35J60.

Key words and phrases. Concentration-compactness principle; variable exponent spaces.

c

2010 Texas State University - San Marcos.

Submitted August 8, 2009. Published October 5, 2010.

1

Finally, as an application of our result, we prove the existence of solutions to the problem

−∆_p(x)u=|u|^q(x)−2u+λ(x)|u|^r(x)−2u in Ω

u= 0 on∂Ω (1.2)

where Ω is a bounded smooth domain inR^N, r(x)< p^∗(x)−δ,q(x)≤p^∗(x) with {q(x) =p^∗(x)} 6=∅.

1.1. Statement of the results. As we already mentioned, the main result of the paper is the extension of Lions concentration - compactness method to the variable exponent case. More precisely, we prove the following result.,

Theorem 1.1. Let q(x)andp(x)be two continuous functions such that 1< inf

x∈Ωp(x)≤sup

x∈Ω

p(x)< n and 1≤q(x)≤p^∗(x) inΩ.

Let {u_j}_j∈N be a weakly convergent sequence in W₀^1,p(x)(Ω) with weak limitu, and such that:

• |∇uj|^p(x)* µweakly-* in the sense of measures.

• |uj|^q(x)−→ν weakly-* in the sense of measures.

Also assume that A = {x ∈ Ω :q(x) = p^∗(x)} is nonempty. Then, for some countable index set I, we have:

ν=|u|^q(x)+X

i∈I

ν_iδ_xi ν_i>0 (1.3) µ≥ |∇u|^p(x)+X

i∈I

µiδx_i µi>0 (1.4) Sν^1/p

∗(x_i)

i ≤µ^1/p(x_i ⁱ⁾ ∀i∈I. (1.5)

where {x_i}_i∈I ⊂ A and S is the best constant in the Gagliardo-Nirenberg-Sobolev inequality for variable exponents, namely

S=S_q(Ω) := inf

φ∈C₀^∞(Ω)

k|∇φ|k_Lp(x)(Ω)

kφk_Lq(x)(Ω)

.

We remark that in Theorem 1.1 is not required the exponentq(x) to be critical everywhere and that the point masses are located in the criticality set A={x∈ Ω :q(x) =p^∗(x)}.

Now, as an application of Theorem 1.1, following the techniques in [11], we prove the existence of solutions to

−∆_p(x)u=|u|^q(x)−2u+λ(x)|u|^r(x)−2u in Ω

u= 0 on∂Ω. (1.6)

In the spirit of [11], we have two types of results, depending onr(x) being smaller or bigger thatp(x). More precisely, we prove the following two theorems.

Theorem 1.2. Letp(x)andq(x)be as in Theorem 1.1 and let r(x)be continuous.

Moreover, assume thatmax_Ωp <min_Ωq andmax_Ωr <min_Ωp. Then, there exists a constant λ1 > 0 depending only on p, q, r, N and Ω such that if λ(x) verifies 0 <inf_x∈Ωλ(x) ≤ kλk_L∞(Ω) < λ1, then there exists infinitely many solutions to (1.6)inW₀^1,p(x)(Ω).

Theorem 1.3. Letp(x)andq(x)be as in Theorem 1.1 and let r(x)be continuous.

Moreover, assume that max_Ωp < min_Ωr and that there exists η > 0 such that r(x)≤p^∗(x)−η in Ω.

Then, there exists λ₀>0 depending only onp, q, r, N andΩ, such that if

x∈Ainfδ

λ(x)> λ0 for someδ >0,

problem (1.6) has at least one nontrivial solution in W₀^1,p(x)(Ω). Here, Aδ is the δ-tubular neighborhood of A, namely

A_δ :=∪_x∈A(B_δ(x)∩Ω).

Organization of this article. After finishing this introduction, in Section 2 we give a very short overview of some properties of variable exponent Sobolev spaces that will be used throughout the paper. In Section 3 we deal with the main result of the paper. Namely the proof of the concentration - compactness principle (The- orem 1.1). In Section 4, we begin analyzing problem (1.6) and prove Theorem 1.3.

Finally, in Section 5, we prove Theorem 1.2.

Comment on a related result. After this paper was written, we found out that a similar result was obtained independently by Yongqiang Fu [10]. Even the techniques in Fu’s work are similar to the ones in this paper (and both are related to the original work by Lions), we want to remark that our results are slightly more general than those in [10]. For instance, we do not requireq(x) to be critical everywhere (as is required in [10]) and we obtain that the delta functions are located in the criticality setA(see Theorem 1.1).

Also, in our application, again as we do not required the source term to be critical everywhere, so the result in [10] is not applicable directly. Moreover, in Theorem 1.3 our approach allows us to consider λ(x) not necessarily a constant and the restriction thatλis large is only needed in anL^∞-norm in the criticality set.

We believe that these improvements are significant and made our result more flexible that those in [10].

2. Results on variable exponent Sobolev spaces The variable exponent Lebesgue spaceL^p(x)(Ω) is defined as

L^p(x)(Ω) ={u∈L¹_loc(Ω) : Z

Ω

|u(x)|^p(x)dx <∞}.

This space is endowed with the norm kuk_Lp(x)(Ω)= inf{λ >0 :

Z

Ω

|u(x)

λ |^p(x)dx≤1}

The variable exponent Sobolev spaceW^1,p(x)(Ω) is defined as

W^1,p(x)(Ω) ={u∈W_loc^1,1(Ω) :u∈L^p(x)(Ω) and|∇u| ∈L^p(x)(Ω)}.

The corresponding norm for this space is

kuk_W1,p(x)(Ω)=kuk_Lp(x)(Ω)+k|∇u|k_Lp(x)(Ω)

Define W₀^1,p(x)(Ω) as the closure ofC₀^∞(Ω) with respect to the W^1,p(x)(Ω) norm.

The spacesL^p(x)(Ω),W^1,p(x)(Ω) andW₀^1,p(x)(Ω) are separable and reflexive Banach spaces when 1<inf_Ωp≤sup_Ωp <∞.

As usual, we denote p⁰(x) = p(x)/(p(x)−1) the conjugate exponent of p(x).

Define

p^∗(x) =

( _{N p(x)}

N−p(x) ifp(x)< N

∞ ifp(x)≥N . The following results are proved in [9].

Proposition 2.1 (H¨older-type inequality). Let f ∈L^p(x)(Ω) and g ∈ L^p0(x)(Ω).

Then the following inequality holds Z

Ω

|f(x)g(x)|dx≤Cpkfk_Lp(x)(Ω)kgk_Lp0(x)(Ω).

Proposition 2.2 (Sobolev embedding). Let p, q∈C(Ω) be such that 1≤q(x)≤ p^∗(x) for all x∈ Ω. Assume moreover that the functions p and q are log-H¨older continuous. Then there is a continuous embedding

W^1,p(x)(Ω),→L^q(x)(Ω).

Moreover, if inf_Ω(p^∗−q)>0 then, the embedding is compact.

Proposition 2.3 (Poincar´e inequality). There is a constant C >0, such that kuk_Lp(x)(Ω)≤Ck|∇u|k_Lp(x)(Ω),

for allu∈W₀^1,p(x)(Ω).

Remark 2.4. By Proposition 2.3, we know thatk|∇u|k_Lp(x)(Ω) andkuk_W1,p(x)(Ω)

are equivalent norms onW₀^1,p(x)(Ω).

In this article, the following notation will be used: Given q: Ω →R bounded, we denote

q⁺:= sup

Ω

q(x), q⁻:= inf

Ω q(x).

The following proposition is also proved in [9] and it will be very useful here.

Proposition 2.5. Set ρ(u) :=R

Ω|u(x)|^p(x)dx. For u,∈L^p(x)(Ω) and {uk}_k∈N⊂ L^p(x)(Ω), we have

u6= 0⇒

kuk_Lp(x)(Ω)=λ⇔ρ(u λ) = 1

. (2.1)

kuk_Lp(x)(Ω)<1(= 1;>1)⇔ρ(u)<1(= 1;>1). (2.2) kuk_Lp(x)(Ω)>1⇒ kuk^p_L⁻_p(x)(Ω)≤ρ(u)≤ kuk^p_L⁺_p(x)(Ω). (2.3) kuk_Lp(x)(Ω)<1⇒ kuk^p_L⁺_p(x)(Ω)≤ρ(u)≤ kuk^p_L⁻_p(x)(Ω). (2.4)

k→∞lim kukk_Lp(x)(Ω)= 0⇔ lim

k→∞ρ(uk) = 0. (2.5)

k→∞lim ku_kk_Lp(x)(Ω)=∞ ⇔ lim

k→∞ρ(u_k) =∞. (2.6)

3. concentration compactness principle

Let{uj}j∈Nbe a bounded sequence inW₀^1,p(x)(Ω) and letq∈C(Ω) be such that q ≤ p^∗ with {x ∈ Ω :q(x) = p^∗(x)} 6= ∅. Then there exists a subsequence, still denoted by{uj}j∈N, such that

• uj* u weakly inW₀^1,p(x)(Ω),

• uj→u strongly inL^r(x)(Ω) ∀1≤r(x)< p^∗(x),

• |uj|^q(x)* ν weakly * in the sense of measures,

• |∇uj|^p(x)* µweakly * in the sense of measures.

Consider φ ∈ C^∞(Ω), from the Poincar´e inequality for variable exponents, we obtain

kφujk_Lq(x)(Ω)S≤ k∇(φuj)k_Lp(x)(Ω). (3.1) On the other hand,

|k∇(φu_j)k_Lp(x)(Ω)− kφ∇u_jk_Lp(x)(Ω)| ≤ ku_j∇φk_Lp(x)(Ω).

We first assume thatu= 0. Then, we observe that the right side of the inequality converges to 0. In fact, if, for instancek|u|^p(x)k_L1(Ω)≥1,

kuj∇φk_Lp(x)(Ω)≤(k∇φk_L^∞_(Ω)+ 1)^p+kujk_Lp(x)(Ω)

≤(k∇φk_L^∞_(Ω)+ 1)^p+k|u|^p(x)k^1/p_L1(Ω)⁻ →0

Now we want to take the limit in (3.1). To do this, we need the following Lemma.

Lemma 3.1. Let{νj}_j∈N, ν be nonnegative, finite Radon measures inΩsuch that νj * ν weakly* in the sense of measures. Then

kφk_Lq(x)

νj (Ω)→ kφk_Lq(x)

ν (Ω) asj→ ∞, for allφ∈C^∞(Ω).

Proof. First, observe that for φ ∈ C^∞(Ω) fixed and for any nonnegative, finite Radon measureµ, the function

hµ(λ) :=

Z

Ω

φ(x) λ

q(x)

dµ

is continuous, decreasing with hµ(0) = +∞ and hµ(+∞) = 0. Hence, if λµ = kφk_Lp(x)

µ (Ω) we have that Z

Ω

φ(x) λ_µ

q(x)

dµ= 1.

Now, letλ=kφk_Lq(x)

ν (Ω)+ε. Hence Z

Ω

φ(x) λ

q(x)dν <1.

Now, asν_j→ν weakly* in the sense of measures, Z

Ω

φ(x) λ

q(x)dνj→ Z

Ω

φ(x) λ

q(x)dν <1.

Therefore, forj large, kφk_Lq(x)

νj (Ω)< λ=kφk_Lq(x) ν (Ω)+ε,

and so

lim sup

j→∞

kφk_Lq(x)

νj (Ω)≤ kφk_Lq(x) ν (Ω). Letλ0:= lim infj→∞kφk_Lq(x)

νj (Ω) and assume thatλ0<kφk_Lq(x) ν (Ω). We can assume thatλ0:= lim_j→∞kφk_Lq(x)

νj (Ω). It is easy to see that fj(x) :=

φ(x) λ_j

q(x)

→f0(x) :=

φ(x) λ₀

q(x)

asj→ ∞ uniformly in Ω and so, asj→ ∞,

1 = Z

Ω

φ(x) λj

q(x)dνj→ Z

Ω

φ(x) λ0

q(x)dν <1,

a contradiction. The proof is completed.

Finally, if we take the limit forj→ ∞in (3.1), by Lemma 3.1, we have kφk_Lq(x)

ν (Ω)S≤ kφk_Lp(x)

µ (Ω) (3.2)

Now we need a lemma that is the key role in the proof of Theorem 1.1.

Lemma 3.2. Let µ, ν be two non-negative and bounded measures on Ω, such that for1≤p(x)< r(x)<∞there exists some constantC >0 such that

kφk_Lr(x)

ν (Ω)≤Ckφk_Lp(x) µ (Ω)

Then, there exist {xj}j∈J⊂Ωand{νj}j∈J⊂(0,∞), such that ν= Σν_iδ_xi

For the proof of the lemma above, we need a couple of preliminary results.

Lemma 3.3. Let ν be a non-negative bounded measure. Assume that there exists δ >0 such that for allA Borelian,ν(A) = 0or ν(A)≥δ. Then, there exist {xi} andνi >0 such that

ν =X νiδx_i

The proof of the above lemma is elementary and is omitted.

Lemma 3.4. Let ν be non-negative and bounded measures, such that kψk_Lr(x)

ν (Ω)≤Ckψk_Lp(x) ν (Ω)

Then there existδ >0 such that for all ABorelian,ν(A) = 0orν(A)≥δ.

Proof. First, observe that ifν(A)≥1, Z

Ω

χ_A(x) ν(A)^p−1

^p(x) dν≤

Z

Ω

χ_A(x) ν(A)^p(x)1

^p(x)

dν = 1.

Thenν(A)^p−1 ≥ kχAk_Lp(x)

ν . On the other hand, Z

Ω

χA(x) ν(A)^r+1

^r(x) dν≥

Z

Ω

χA(x)

ν(A) dν = 1.

Thenν(A)^r+1 ≤ kχAk_Lr(x)

ν . So we conclude that ν(A)^r+1 ≤Cν(A)^p−1 .

Now, ifν(A)<1, we obtain

ν(A)^r−1 ≤Cν(A)^p+1 . Combining all these facts, we arrive at

min{ν(A)^r−1 , ν(A)^r+1 } ≤Cmax{ν(A)^p−1 , ν(A)^p+1 }.

Now, ifν(A)≤1, we have

ν(A)^r−1 ≤Cν(A)^p+1 . Then,ν(A) = 0 or

ν(A)≥(1 C)

p+r− r− −p+. Finally,

ν(A)≥min{(1 C)

p+r− r− −p+,1}

This completes the proof.

In the rest of the proofs we will use the following notation: Given a Radon measureµin Ω and a funcion f ∈L¹_µ(Ω) we denote the restriction ofµtof by

µbf(E) :=

Z

E

f dµ.

Proof of Lemma 3.2. By reverse H¨older inequality (3.2), the measureνis absolutely continuous with respect toµ. As consequence there existsf ∈L¹_µ(Ω),f ≥0, such thatν =µbf. Also by (3.2), we have

min

ν(A)^r−1 , ν(A)^r+1 ≤Cmax

µ(A)^p−1 , µ(A)^p+1

for any Borel set A ⊂ Ω. In particular, f ∈ L^∞_µ(Ω). On the other hand the Lebesgue decomposition ofµwith respect to ν gives us

µ=νbg+σ, where g∈L¹_ν(Ω), g≥0 andσis a bounded positive measure, singular with respect toν.

Now consider (3.2) applying the test function φ=g^r(x)−p(x)1 χ_{g≤n}ψ.

We obtain

kg^r(x)−p(x)1 χ_{g≤n}ψk_Lr(x) ν

≤Ckg^r(x)−p(x)1 χ_{g≤n}ψk_Lp(x) µ

=Ckg^r(x)−p(x)1 χ_{g≤n}ψk_Lp(x) gdν+dσ

≤Ckgp(x)(r(x)−p(x))^r(x) χ_{g≤n}ψk_Lp(x)

ν +Ckg^r(x)−p(x)1 χ_{g≤n}ψk_Lp(x) σ

Sinceσ⊥ν, we have

kg^r(x)−p(x)1 χ_{g≤n}ψk_Lr(x)

ν ≤Ckgp(x)(r(x)−p(x))^r(x) χ_{g≤n}ψk_Lp(x) ν

Hence callingdνn=g

r(x)

(r(x)−p(x))χg≤ndνthe following reverse H¨older inequality holds kψk_Lr(x)

νn ≤Ckψk_Lp(x) νn .

By Lemma 3.3 and Lemma 3.4, there exists xⁿ_i and K_iⁿ > 0 such that νn = P

i∈IK_iⁿδxⁿ_i. On the other hand,νn%g^{r(x)−p(x)r(x)} ν. Then, the pointsxⁿ_i are in fact independent ofn, and there will denoted byxi, and the numbersK_iⁿ are monotone inn. Then, we have

g

r(x)

r(x)−p(x)ν=X

i∈I

Kiδx_i

whereKi=g

r(xi)

r(xi)−p(xi)(xi)ν(xi). This finishes the proof.

The following Lemma follows exactly as in the constant exponent case and the proof is omitted.

Lemma 3.5. Let fn →f a.e andfn* f in L^p(x)(Ω) then

n→∞lim Z

Ω

|fn|^p(x)dx− Z

Ω

|f−fn|^p(x)dx

= Z

Ω

|f|^p(x)dx Now we are in position to prove the main results.

Proof of Theorem 1.1. Given anyφ∈C^∞(Ω), we writev_j=u_j−uand by Lemma 3.5, we have

j→∞lim Z

Ω

|φ|^q(x)|uj|^q(x)− Z

Ω

|φ|^q(x)|vj|^q(x)dx

= Z

Ω

|φ|^q(x)|u|^q(x)dx.

On the other hand, by reverse H¨older inequality (3.2) and Lemma 3.2, taking limits we obtain the representation

ν =|u|^q(x)+X

j∈I

νjδx_j

Let us now show that the points x_j actually belong to the critical setA. In fact, assume by contradiction that x₁ ∈ Ω\ A. Let B = B(x₁, r) ⊂⊂ Ω− A. Then q(x) < p^∗(x)−δ for some δ > 0 in B and, by Proposition 2.2, The embedding W^1,p(x)(B),→L^q(x)(B) is compact. Therefore, u_j →u strongly in L^q(x)(B) and so |u_j|^q(x) → |u|^q(x) strongly inL¹(B). This is a contradiction to our assumption thatx₁∈B.

Now we proceed with the proof. Applying (3.1) toφu_j and taking into account thatu_j→uinL^p(x)(Ω), we have

Skφk_Lq(x)

ν (Ω)≤ kφk_Lp(x)

µ (Ω)+k(∇φ)uk_Lp(x)(Ω).

Consider φ ∈C_c^∞(Rⁿ) such that 0 ≤φ ≤1,φ(0) = 1 and supported in the unit ball ofRⁿ. Fixedj∈I, we considerε >0 be arbitrary.

We denote byφε,j(x) :=ε⁻ⁿφ((x−xj)/ε). By decomposition ofν, we have:

ρν(φi₀,ε) :=

Z

Ω

|φi₀,ε|^q(x)dν

= Z

Ω

|φi₀,ε|^q(x)|u|^q(x)dx+X

i∈I

νiφi₀,ε(xi)^q(xi)≥νi₀. For the rest of this article, we will denote

q_i,ε⁺ := sup

B_ε(x_i)

q(x), q_i,ε⁻ := inf

B_ε(x_i)q(x), p⁺_i,ε:= sup

B_ε(x_i)

p(x), p⁻_i,ε := inf

Bε(xi)

p(x).

Ifρν(φi₀,ε)<1 then kφi₀,εk_Lq(x)

ν (Ω)=kφi₀,εk_Lq(x)

ν (Bε(xi0)) ≥ρν(φi₀,ε)^1/q−i,ε ≥ν^1/q

− i,ε

i₀ . Analogously, ifρ_ν(φ_i0,ε)>1, then

kφi0,εk_Lq(x)

ν (Ω)≥ν^1/q

+ i,ε

i0 . Then

min{ν

1 q+ i,ε

i , ν

1 q− i,ε

i }S ≤ kφi,εk_Lp(x)

µ (Ω)+k(∇φi,ε)uk_Lp(x)(Ω). By Proposition 2.5,

k(∇φi,ε)uk_Lp(x)(Ω)≤max{ρ((∇φi,ε)u)^1/p−;ρ((∇φi,ε)u)^1/p+}.

Then, by H¨older inequality, we have ρ((∇φi,ε)u) =

Z

Ω

|∇φi,ε|^p(x)|u|^p(x)dx

≤ k|u|^p(x)k_Lα(x)(B_ε(x_i))k|∇φi,ε|^p(x)k_Lα0(x)(B_ε(x_i)), whereα(x) =n/(n−p(x)) andα⁰(x) =n/p(x).

Moreover, using that∇φi,ε=∇φ ^x−x_ε ⁱ₁

ε, we obtain

k|∇φ_i,ε|^p(x)k_Lα0(x)(Bε(xi))≤max{ρ(|∇φ_i,ε|^p(x))^p+/n;ρ(|∇φ_i,ε|^p(x))^p−/n}, and

ρ(|∇φi,ε|^p(x)) = Z

B_ε(x_i)

|∇φi,ε|ⁿdx

= Z

Bε(xi)

|∇φ(x−xi

ε )|ⁿ 1 εⁿ dx

= Z

B₁(0)

|∇φ(y)|ⁿdy.

Then∇φi,εu→0 strongly inL^p(x)(Ω). On the other hand, Z

Ω

|φi,ε|^p(x)dµ≤µ(Bε(xi)).

Therefore,

kφi,εk_Lp(x)(Ω)=kφi,εk_Lp(x)(B_ε(x_i))

≤max{ρ_µ(φ_i,ε)^1/p+i,ε, ρ_µ(φ_i,ε)^1/p−i,ε}

≤max{µ(Bε(xi))^1/p+i,ε, µ(Bε(xi))^1/p−i,ε}, so we obtain,

Smin{ν

1 q+ i,ε

i , ν

1 q− i,ε

i } ≤max{µ(Bε(xi))^1/p+i,ε, µ(Bε(xi))

1 p−

i,ε}.

Aspandqare continuous functions and asq(x_i) =p^∗(x_i), letting ε→0, we get Sν^1/p

∗(x_i)

i ≤µ^1/p(x_i ⁱ⁾, whereµi:= lim_ε→0µ(Bε(xi)).

Finally, we show thatµ≥ |∇u|^p(x)+ Σµiδx_i. In fact, we have thatµ≥P µiδx_i. On the other hand u_j * u weakly in W₀^1,p(x)(Ω) then ∇uj * ∇u weakly in

L^p(x)(U) for allU ⊂Ω. By weakly lower semi continuity of norm we obtain that dµ ≥ |∇u|^p(x)dx and, as |∇u|^p(x) is orthogonal to µ1, we conclude the desired

result. This completes the proof.

4. Applications

In this section, we apply Theorem 1.1 to study the existence of nontrivial solu- tions of the problem

−∆_p(x)u=|u|^q(x)−2u+λ(x)|u|^r(x)−2u in Ω,

u= 0 on∂Ω, (4.1)

where r(x)< p^∗(x)−ε, q(x) ≤p^∗(x) and A= {x∈ Ω :q(x) = p^∗(x)} 6=∅. We define Aδ := S

x∈A(Bδ(x)∩Ω) = {x ∈ Ω : dist(x,A) < δ}. The ideas for this application follow those in [11].

For (weak) solutions of (4.1) we understand critical points of the functional F(u) =

Z

Ω

|∇u|^p(x)

p(x) −|u|^q(x)

q(x) −λ(x)|u|^r(x) r(x) dx

4.1. Proof of Theorem 1.3. We begin by proving the Palais-Smale condition for the functionalF, below certain level of energy.

Lemma 4.1. Assume that r ≤q. Let {uj}j∈N ⊂W₀^1,p(x)(Ω) a Palais-Smale se- quence then {uj}_j∈N is bounded inW₀^1,p(x)(Ω).

Proof. By definitionF(u_j)→candF⁰(u_j)→0. Now, we have c+ 1≥ F(u_j) =F(u_j)− 1

r−hF⁰(u_j), u_ji+ 1

r−hF⁰(u_j), u_ji, where

hF⁰(uj), uji= Z

Ω

|∇uj|^p(x)− |uj|^q(x)−λ(x)|uj|^r(x) dx.

Then, ifr(x)≤q(x), we conclude that c+ 1≥ 1

p+− 1 r−

Z

Ω

|∇u_j|^p(x)dx− 1

r−|hF⁰(u_j), u_ji|.

We can assume thatkujk_W1,p(x)

0 (Ω)≥1. AskF⁰(uj)kis bounded we have that c+ 1≥ 1

p+− 1 r−

kujk^p−

W₀^1,p(x)(Ω)− C

r−kujk_W1,p(x)

0 (Ω).

We deduce thatuj is bounded. This completes the proof.

From the fact that{u_j}_j∈Nis a Palais-Smale sequence it follows, by Lemma 4.1, that{uj}j∈Nis bounded inW₀^1,p(x)(Ω). Hence, by Theorem 1.1, we have

|uj|^q(x)* ν =|u|^q(x)+X

i∈I

ν_iδ_xi ν_i>0, (4.2)

|∇u_j|^p(x)* µ≥ |∇u|^p(x)+X

i∈I

µ_iδ_xi µ_i>0, (4.3) Sν_i^1/p∗(xi)≤µ^1/p(x_i ⁱ⁾. (4.4) Note that ifI=∅thenu_j→ustrongly inL^q(x)(Ω). We know that{xi}i∈I ⊂ A.

Let us show that ifc < _p¹+ − ¹

q⁻_A

Sⁿ and {uj}j∈N is a Palais-Smale sequence, with energy levelc, thenI=∅. In fact, suppose thatI6=∅. Then letφ∈C₀^∞(Rⁿ) with support in the unit ball of Rⁿ. Consider, as in the previous section, the rescaled functionsφi,ε(x) =φ(^x−x_ε ⁱ).

AsF⁰(uj)→0 in (W₀^1,p(x)(Ω))⁰, we obtain that

j→∞limhF⁰(u_j), φ_i,εu_ji= 0.

On the other hand, hF⁰(uj), φi,εuji=

Z

Ω

|∇uj|^p(x)−2∇uj∇(φi,εuj)−λ(x)|uj|^r(x)φi,ε− |uj|^q(x)φi,εdx Then, passing to the limit asj→ ∞, we obtain

0 = lim

j→∞

Z

Ω

|∇uj|^p(x)−2∇uj∇(φi,ε)ujdx +

Z

Ω

φi,εdµ− Z

Ω

φi,εdν− Z

Ω

λ(x)|u|^r(x)φi,εdx.

By H¨older inequality, it is easy to check that

j→∞lim Z

Ω

|∇u_j|^p(x)−2∇u_j∇(φ_i,ε)u_jdx= 0.

On the other hand,

ε→0lim Z

Ω

φ_i,εdµ=µ_iφ(0), lim

ε→0

Z

Ω

φ_i,εdν =ν_iφ(0),lim

ε→0

Z

Ω

λ(x)|u|^r(x)φ_i,εdx= 0.

So, we conclude that (µ_i−ν_i)φ(0) = 0; i.e.,µ_i=ν_i. Then Sν^1/p

∗(x_i)

i ≤ν_i^1/p(xi); so it is clear thatνi= 0 orSⁿ≤νi.

On the other hand, asr⁻> p⁺, c= lim

j→∞F(uj) = lim

j→∞F(uj)− 1

p+hF⁰(uj), uji

= lim

j→∞

Z

Ω

1 p(x)− 1

p+

|∇uj|^p(x)dx+ Z

Ω

1 p+− 1

q(x)

|uj|^q(x)dx +λ

Z

Ω

1 p+− 1

r(x)

|uj|^r(x)dx

≥ lim

j→∞

Z

Ω

1 p+− 1

q(x)

|uj|^q(x)dx

≥ lim

j→∞

Z

Aδ

1 p+− 1

q(x)

|uj|^q(x)dx

≥ lim

j→∞

Z

Aδ

1 p+− 1

q_A⁻

δ

|uj|^q(x)dx . However,

j→∞lim Z

Aδ

1 p+− 1

q_A⁻

δ

|uj|^q(x)dx= 1 p+− 1

q_A⁻

δ

Z

Aδ

|u|^q(x)dx+X

j∈I

νj

≥ 1 p+− 1

q_A⁻

δ

ν_i

≥ 1 p+− 1

q_A⁻

δ

Sⁿ. Asδis positive and arbitrary, andqis continuous, we have

c≥ 1 p+− 1

q⁻_A Sⁿ. Therefore, if

c < 1 p+− 1

q⁻_A Sⁿ, the index setI is empty.

Now we are ready to prove the Palais-Smale condition below levelc.

Theorem 4.2. Let {uj}_j∈N⊂W₀^1,p(x)(Ω)be a Palais-Smale sequence, with energy level c. If c < _p+¹ − ¹

q_A⁻

Sⁿ, then there exist u ∈ W₀^1,p(x)(Ω) and {ujk}k∈N ⊂ {uj}_j∈N a subsequence such thatuj_k→ustrongly in W₀^1,p(x)(Ω).

Proof. We have that {uj}_j∈N is bounded. Then, for a subsequence that we still denote {uj}_j∈N, uj → u strongly in L^q(x)(Ω). We define F⁰(uj) := φj. By the Palais-Smale condition, with energy level c, we haveφ_j→0 in (W₀^1,p(x)(Ω))⁰.

By definitionhF⁰(uj), zi=hφj, zifor allz∈W₀^1,p(x)(Ω); i.e., Z

Ω

|∇u_j|^p(x)−2∇u_j∇z dx− Z

Ω

|u_j|^q(x)−2u_jz dx− Z

Ω

λ(x)|u_j|^r(x)−2u_jz dx=hφ_j, zi.

Then,uj is a weak solution of the following equation.

−∆_p(x)u_j =|u_j|^q(x)−2u_j+λ(x)|u_j|^r(x)−2u_j+φ_j=:f_j in Ω,

uj= 0 on∂Ω. (4.5)

We defineT: (W₀^1,p(x)(Ω))⁰→W₀^1,p(x)(Ω),T(f) :=uwhereuis the weak solution of the equation

−∆_p(x)u=f in Ω,

u= 0 on∂Ω. (4.6)

ThenT is a continuous invertible operator.

It is sufficient to show thatfjconverges in (W₀^1,p(x)(Ω))⁰. We only need to prove that|u_j|^q(x)−2u_j → |u|^q(x)−2ustrongly in (W₀^1,p(x)(Ω))⁰. In fact,

h|uj|^q(x)−2uj− |u|^q(x)−2u, ψi= Z

Ω

(|uj|^q(x)−2uj− |u|^q(x)−2u)ψ dx

≤ kψk_Lq(x)(Ω)k(|u_j|^q(x)−2u_j− |u|^q(x)−2u)k_Lq0(x)(Ω). Therefore,

k(|uj|^q(x)−2uj− |u|^q(x)−2u)k_(W1,p(x) 0 (Ω))⁰

= sup

ψ∈W1,p(x)

0 (Ω)

kψk W1,p(x)

0 (Ω)=1

Z

Ω

(|uj|^q(x)−2uj− |u|^q(x)−2u)ψ dx

≤ k(|uj|^q(x)−2u_j− |u|^q(x)−2u)k_Lq0(x)(Ω)

and now, by the Dominated Convergence Theorem this last term approaches zero

asj → ∞. The proof is complete.

We are now in position to prove Theorem 1.3.

Proof of Theorem 1.3. In view of the previous result, we seek for critical values below level c. For that purpose, we want to use the Mountain Pass Theorem.

Hence we have to check the following condition:

(1) There exist constants R, r > 0 such that when kuk_W1,p(x)(Ω) = R, then F(u)> r.

(2) There exist v0∈W^1,p(x)(Ω) such thatF(v0)< r.

Let us first check (1). We suppose that k|∇u|k_Lp(x)(Ω) ≤ 1 and kuk_Lp(x)(Ω) ≤1.

The other cases can be treated similarly.

By Poincar´e inequality (Proposition 3.1), we have Z

Ω

|∇u|^p(x)

p(x) −|u|^q(x)

q(x) −λ(x)|u|^r(x) r(x) dx

≥ 1 p+

Z

Ω

|∇u|^p(x)dx− 1 q−

Z

Ω

|u|^q(x)dx−kλk∞

r−

Z

Ω

|u|^r(x)dx

≥ 1

p+k|∇u|k^p+− 1

q−kuk^q−_Lq(x)(Ω)−kλk∞

r− kuk^r−_Lr(x)(Ω)

≥ 1

p+k|∇u|k^p+− C

q−k|∇u|k^q−_Lp(x)(Ω)−Ckλk_∞

r− k|∇u|k^r−_Lp(x)(Ω).

Letg(t) = _p+¹ t^p+−_q−^Ct^q−−^Ckλk_r−^∞t^r−, then it is easy to check thatg(R)> r for someR, r >0. This proves (1).

Now (2) is immediate as for a fixedw∈W₀^1,p(x)(Ω) we have

t→∞lim F(tw) =−∞.

Now the candidate for critical value according to the Mountain Pass Theorem is c= inf

g∈C sup

t∈[0,1]

F(g(t)),

whereC={g: [0,1]→W₀^1,p(x)(Ω) : gcontinuous andg(0) = 0, g(1) =v0}.

We will show that, if inf_x∈Aδλ(x) is big enough for someδ >0 thenc < _p+¹ −

1 q_A⁻

Sⁿ and so the local Palais-Smale condition (Theorem 4.2) can be applied. We fixw∈W₀^1,p(x)(Ω). Then, ift <1, we have

F(tw)≤ Z

Ω

t^p(x)|∇w|^p(x)

p− −t^q(x)|w|^q(x)

q+ −λ(x)t^r(x)|w|^r(x) r+ dx

≤ t^p−

p−

Z

Ω

|∇w|^p(x)dx−t^r+

r+

Z

Ω

λ(x)|w|^r(x)dx

≤ t^p−

p−

Z

Ω

|∇w|^p(x)dx−t^r+

r+

Z

A_δ

λ(x)|w|^r(x)dx

≤ t^p−

p−

Z

Ω

|∇w|^p(x)dx−t^r+

r+

Z

Aδ

( inf

x∈Aδ

λ(x))|w|^r(x)dx

We defineg(t) := ^t_p−^p−a₁−(inf_x∈Aδλ(x))^t_r+^r+a₃, wherea₁ anda₂ are given bya₁= k|∇w|^p(x)kL¹(Ω)and a₃=k|w|^r(x)kL¹(Aδ).

The maximum ofg is attained attλ= _(inf ^a1

x∈Aδλ(x))a3

_r+−p−¹

. So, we conclude that there existsλ0>0 such that if (infx∈A_δλ(x))≥λ0 then

F(tw)< 1 p+− 1

q⁻_A Sⁿ

This completes the proof.

Remark 4.3. Observe that if λ(x) is continuous it suffices to assume thatλ(x) is large in thecriticality setA.

4.2. Proof of Theorem 1.2. Now it remains to prove Theorem 1.2. So we begin by checking the Palais-Smale condition for this case.

Lemma 4.4. Let {uj}_j∈N ⊂ W₀^1,p(x)(Ω) be a Palais-Smale sequence for F then {uj}j∈N is bounded.

Proof. Let{uj}_j∈N⊂W₀^1,p(x)(Ω) be a Palais-Smale sequence; that is,F(uj)→ c andF⁰(uj)→0. Therefore there exists a sequenceεj →0 such that

|F⁰(u_j)w| ≤ε_jkwk_W1,p(x)

0 (Ω) for allw∈W₀^1,p(x)(Ω).

Now we have c+ 1≥ F(u_j)− 1

q⁻F⁰(u_j)u_j+ 1

q⁻F⁰(u_j)u_j

≥ 1 p⁺ − 1

q⁻

Z

Ω

|∇uj|^p(x)dx+ Z

Ω

λ(x) q⁻ −λ(x)

r⁻

|uj|^r(x)dx+ 1

q⁻F⁰(uj)uj

We can assume thatk|∇u_j|k_Lp(x)(Ω)>1. Then we have, by Proposition 2.5 and by Poincar´e inequality,

c+ 1≥ 1 p⁺ − 1

q⁻

k|∇uj|k^p_L⁻_p(x)(Ω)+kλk∞

1 q⁻ − 1

r⁻

kujk^r_L⁺r(x)(Ω)

− 1

q⁻kujk_W1,p(x) 0 (Ω)εj

≥ 1 p⁺ − 1

q⁻

k|∇uj|k^p_L⁻p(x)(Ω)+kλk∞

1 q⁻ − 1

r⁻

Ck|∇uj|k^r_L⁺p(x)(Ω)

− 1

q⁻ku_jk_W1,p(x)

0 (Ω)

from where it follows that kujk_W1,p(x)

0 (Ω) is bounded (recall that p⁺ ≤ q⁻ and

r⁺< p⁻).

Let {uj}j∈N be a Palais-Smale sequence for F. Therefore, by the previous Lemma, it follows that{uj}_j∈N is bounded inW₀^1,p(x)(Ω).

Then, by Theorem 1.1 we can assume that there exist two measuresµ, ν and a functionu∈W₀^1,p(x)(Ω) such that

u_j* u weakly inW₀^1,p(x)(Ω), (4.7)

|∇uj|^p(x)* µ weakly in the sense of measures, (4.8)

|uj|^q(x)* ν weakly in the sense of measures, (4.9) ν =|u|^q(x)+X

i∈I

ν_iδ_xi, (4.10)

µ≥ |∇u|^p(x)+X

i∈I

µ_iδ_xi, (4.11)

Sν_i^1/p∗(xi)≤µ^1/p(x_i ⁱ⁾. (4.12) As before, assume that I 6= ∅. Now the proof follows exactly as in the previous case, until we get to

c≥ 1 p⁺ − 1

q⁻

Z

Ω

|u|^q(x)dx+ 1 p⁺ − 1

q⁻

Sⁿ+kλk_L∞(Ω)

1 p⁺ − 1

r⁻

Z

Ω

|u|^r(x)dx.

Applying now H¨older inequality, we find c≥ 1

p⁺ − 1 q⁻

Z

Ω

|u|^q(x)dx+ 1 p⁺ − 1

q⁻ Sⁿ +kλk_L^∞_(Ω) 1

p⁺ − 1 r⁻

k|u|^r(x)k_Lq(x)/r(x)(Ω)|Ω| ^q

+ q− −r+. Ifk|u|^r(x)k_Lq(x)/r(x)(Ω)≥1, we have

c≥c1k|u|^r(x)k^(q/r)_Lq(x)/r(x)⁻ _(Ω)+c3− kλk_L∞(Ω)c2k|u|^r(x)k_Lq(x)/r(x)(Ω),

so, iff1(x) :=c1x^(q/r)−− kλk_L^∞_(Ω)c2x, this function reaches its absolute minimum atx₀= ^kλk_c^L∞(Ω)c2

1(q/r)⁻

_{(q/r)− −1}¹ .

On the other hand, ifk|u|^r(x)k_Lq(x)/r(x)(Ω)<1, then

c≥c₁k|u|^r(x)k^(q/r)_Lq(x)/r(x)⁺ (Ω)+c₃− kλkL^∞(Ω)c₂kuk_Lq(x)/r(x)(Ω),

so, iff2(x) =c1x^(q/r)+− kλk_L∞(Ω)c2x, this function reaches its absolute minimum atx0= ^kλk_c^L∞(Ω)c2

1(q/r)⁺

_(q/r)+−1¹ . Then c≥ 1

p+− 1 q⁻

Sⁿ+Kmin{kλk

(q/r)− (q/r)− −1

L^∞(Ω) ,kλk

(q/r)+

(q/r)+−1

L^∞(Ω) },

which contradicts our hypothesis. Therefore I = ∅ and so uj → u strongly in L^q(x)(Ω).

With these preliminaries the Palais-Smale condition can now be easily checked.

Lemma 4.5. Let(u_j)⊂W₀^1,p(x)(Ω)be a Palais-Smale sequence forF, with energy level c. There exists a constant K depending only on p, q, r and Ω such that, if c < _p+¹ −_q¹−

Sⁿ+Kmin{kλk

(q/r)− (q/r)− −1

L^∞(Ω) ,kλk

(q/r)+

(q/r)+−1

L^∞(Ω) }, then there exists a subsequence {uj_k}_k∈N⊂ {uj}_j∈N that converges strongly inW₀^1,p(x)(Ω).

The proof of the above lemma follows by the continuity of the solution operator as in Theorem 4.2.

Assume now that k|∇u|k_Lp(x)(Ω) ≤ 1. Then, applying Poincar´e inequality, we have

F(u)≥ 1

p⁺k|∇u|k^p_L⁺_p(x)(Ω)− 1

q⁻kuk^q_L⁻_q(x)(Ω)−kλk_L∞(Ω)

r⁻ kuk^r_L⁻r(x)(Ω)

≥ 1

p⁺k|∇u|k^p_L⁺_p(x)(Ω)− C

q⁻k|∇u|k^q_L⁻_p(x)(Ω)−kλk_L^∞_(Ω)C

r⁻ k|∇u|k^r_L⁻p(x)(Ω)

=:J₁(k|∇u|k_Lp(x)(Ω)),