Existence and Multiplicity of Solutions for Nonlocal ... · Existence and Multiplicity of Solutions...

Existence and Multiplicity of Solutions for Nonlocal Neumann

Problem with Non-Standard Growth

Francisco Julio S.A. Correa ∗

Universidade Federal de Campina GrandeCentro de Ciencias e Tecnologia

Unidade Academica de MatematicaCEP:58.109-970, Campina Grande - PB - Brazil

E-mail: [email protected]&

Augusto Cesar dos Reis CostaUniversidade Federal do Para

Instituto de Ciencias Exatas e NaturaisFaculdade de Matematica

CEP:66.075-110, Belem - PA - BrazilE-mail: [email protected]

June 22, 2015

Abstract

In this paper we are concerning with questions of existence of solution for anonlocal and non-homogeneous Neumann boundary value problems involving thep(x)-Laplacian in which the non-linear terms have critical growth. The main toolswe will use are the generalized Sobolev spaces and the Mountain Pass Theorem.

MSC: 35J60; 35J70; 58E05Keywords: nonlocal problem; Neumann boundary conditions; trace; Sobolev spaceswith variable exponent; critical exponent.

∗Partially supported by CNPq-Brazil under Grant 301807/2013-2.

1

1 Introduction

In this paper we are going to study questions of existence of solutions for the nonlocaland non-homogeneous equation, with critical growth and Neumann boundary condi-tions, given by

M

(∫Ω

1

p(x)(|∇u|p(x) + |u|p(x))dx

)(−div(|∇u|p(x)−2∇u) + |u|p(x)−2u)

= λf(x, u)[∫

ΩF (x, u)dx

]rin Ω

M

(∫Ω

1

p(x)(|∇u|p(x) + |u|p(x))dx

)|∇u|p(x)−2∂u

∂ν

= γg(x, u)[∫∂ΩG(x, u)dS

]κon ∂Ω,

, (1.1)

where Ω ⊂ IRN is a bounded smooth domain of IRN , p ∈ C(Ω), f : Ω×IR→ IR, g : ∂Ω×IR → IR and M : IR+ → IR+ are continuous functions enjoying some conditions which

will be stated later, F (x, u) =∫ u

0f(x, ξ)dξ, G(x, u) =

∫ u

0g(x, y)dy,

∂u

∂νis the outer

unit normal derivative, λ, r, γ, κ are real parameters, and ∆p(x) is the p(x)-Laplacianoperator, that is,

∆p(x)u =N∑i=1

∂

∂xi

(|∇u|p(x)−2 ∂u

∂xi

), 1 < p(x) < N.

Bonder and Silva [3], in 2010, extended the concentration-compactness principle byLions [16] for variable exponent spaces and applied the result to the Dirichlet probleminvolving the p(x)-Laplacian with subcritical and critical growth and showed the exis-tence and multiplicity of solutions via Mountain Pass Theorem, Truncation argumentand Krasnoselskii genus, inspired in Azorero and Peral [2], who solved the problem forthe p-Laplacian. It is important to note that there exists similar result due to Bonderand Silva [3], on concentration-compactness principle, obtained by Fu [13]. Liang andZhang [15], in 2011, showed multiplicity of solutions to the Neumann problem involv-ing the p(x)-Laplacian, with critical growth, via truncation argument. Guo and Zhao[14], in 2012, showed existence and multiplicity of solutions for the nonlocal Neumannproblem involving the p(x)-Laplacian, with subcritical growth, using Mountain PassTheorem and Fountain Theorem. Bonder, Saintier and Silva [4], in 2013, presentedthe concentration-compactness principle for variable exponent in the trace case and in[5] showed existence of solution for the Neumann problem with critical growth on theboundary of the domain, via Mountain Pass Theorem.

In this article, we discuss existence of solutions for the nonlocal problem (1.1) withcritical growth on the domain and on its boundary. We study the nonlocal condition

2

for the two following important classes of functions: M(τ) = a+ bτ η with a ≥ 0, b > 0,η ≥ 1 and m0 ≤ M(τ) ≤ m1, where m0 and m1 are positive constants. Note that theoriginal Kirchhoff term is included in our analysis.

We will study the problem with the following critical Sobolev exponents

p∗(x) =Np(x)

N − p(x)and p∗(x) =

(N − 1)p(x)

N − p(x), (1.2)

where p∗ is critical exponent from the point of view of the trace.Problems in the form (1.1) are associated with the energy functional

Jλ,γ(u) = M

(∫Ω

1

p(x)(|∇u|p(x) + |u|p(x))dx

)− λ

r + 1

[∫ΩF (x, u)dx

]r+1

− γ

κ+ 1

[∫∂ΩG(x, u)dS

]κ+1(1.3)

for all u ∈ W 1,p(x)(Ω), where M(t) =∫ t

0M(s)dS, dS denotes the boundary measure,

and W 1,p(x)(Ω) is the generalized Lebesgue-Sobolev space whose precise definition andproperties will be established in section 2.

Depending on the behavior of the functions p, q the above functional is differentiableand its Frechet-derivative is given by

J ′λ,γ(u)v = M

(∫Ω

1

p(x)(|∇u|p(x) + |u|p(x))dx

)∫Ω

(|∇u|p(x)−2∇u∇v + |u|p(x)−2uv

)dx

−λ[∫

ΩF (x, u)dx

]r ∫Ωf(x, u)vdx− γ

[∫∂ΩG(x, u)dS

]κ ∫∂Ωg(x, u)vdS

(1.4)for all u, v ∈ W 1,p(x)(Ω). So, u ∈ W 1,p(x)(Ω) is a weak solution of problem (1.1) if, andonly if, u is a critical point of Jλ,γ.

Theorem 1.1

(i) Assume κ = 0, g(x, u) = |u|q(x)−2u, q : ∂Ω → [1,∞) and A := x ∈ ∂Ω : q(x) =p∗(x) 6= ∅ and M(τ) = a + bτ η, with a ≥ 0, b > 0, η ≥ 1. Moreover, assume theexistence of functions p(x), q(x), β(x) ∈ C+(Ω), see section 2, positive constants A1, A2

such that A1tβ(x)−1 ≤ f(x, t) ≤ A2t

β(x)−1 for all t ≥ 0 and for all x ∈ Ω, with f(x, t) =

0 for all t < 0. Furthermore, (η + 1)p+ < β−(r + 1) < q− and(η + 1)(p+)η+1

(p−)η<(

A1

A2

)r+1 (β−)r+1(r + 1)

(β+)r. Then there exists λ0 > 0 such that for all λ > λ0 and for all

γ > 0 there exists a nontrivial solution to (1.1).

3

(ii) Assume κ = 0, g(x, u) = |u|q(x)−2u, q : ∂Ω → [1,∞) and A := x ∈ ∂Ω : q(x) =p∗(x) 6= ∅. Moreover, assume the existence of functions p(x), q(x), β(x) ∈ C+(Ω), seesection 2, positive constants A1, A2 such that A1t

β(x)−1 ≤ f(x, t) ≤ A2tβ(x)−1 for all

t ≥ 0 and for all x ∈ Ω, with f(x, t) = 0 for all t < 0. Furthermore, assume there exists

0 < m0 and m1 such that m0 ≤ M(τ) ≤ m1, withm1p

+

m0

<(A1

A2

)r+1 (β−)r+1(r + 1)

(β+)r

and p+ < β−(r+ 1) < q−. Then there exists λ1 > 0 such that for all λ > λ1 and for allγ > 0 there exists a nontrivial solution to (1.1).

(iii) Suppose r = 0, f(x, u) = |u|q(x)−2u, q : Ω → [1,∞) and A := x ∈ Ω : q(x) =p∗(x) 6= ∅. Moreover, assume the existence of functions p(x), q(x), β(x) ∈ C+(∂Ω),positive constants A1, A2 such that A1t

β(x)−1 ≤ g(x, t) ≤ A2tβ(x)−1 for all t ≥ 0 and for

all x ∈ ∂Ω, with g(x, t) = 0 for all t < 0. Furthermore, assume there exists 0 < m0

and m1 such that m0 ≤ M(τ) ≤ m1, withm1p

+

m0

<(A1

A2

)κ+1 (β−)κ+1(κ+ 1)

(β+)κand

p+ < β−(κ + 1) < q−. Then there exists γ1 > 0 such that for all γ > γ1 and for allλ > 0 there exists a nontrivial solution to (1.1).

(iv) Suppose r = 0, f(x, u) = |u|q(x)−2u, q : Ω → [1,∞), A := x ∈ Ω : q(x) =p∗(x) 6= ∅ and M(τ) = a + bτ η, com a ≥ 0, b > 0, τ ≥ 0, and η ≥ 1. Moreover,assume the existence of functions p(x), q(x), β(x) ∈ C+(∂Ω), positive constants A1, A2

such that A1tβ(x)−1 ≤ g(x, t) ≤ A2t

β(x)−1 for all t ≥ 0 and for all x ∈ ∂Ω, withg(x, t) = 0 for all t < 0. Furthermore, assume (η + 1)p+ < β−(κ + 1) < q− and(η + 1)(p+)η+1

(p−)η<(A1

A2

)κ+1 (β−)κ+1(κ+ 1)

(β+)κ. Then there exists γ2 > 0 such that for all

γ > γ2 and for all λ > 0 there exists a nontrivial solution to (1.1).

We should point out that the novelty in the present paper is the appearance of

the integral terms,[∫

ΩF (x, u)dx

]rand

[∫∂ΩG(x, u)dS

]κ, the critical growth on Neu-

mann boundary conditions, the use of the p(x)-Laplacian and the generalized Lebesgue-Sobolev spaces.

We should point out that some ideas contained in this paper were inspired by thearticles Correa and Figueiredo [[6], [7]], Fan [[8], [9]], Fu [13], Liang and Zhang [15],Mihailescu and Radulescu [17] and Yao [20].

This paper is organized as follows: In section 2 we present some preliminaries onthe variable exponent spaces. In section 3, we proof of our main result.

2 Preliminaries on variable exponent spaces

First of all, we set

C+(Ω) :=h;h ∈ C(Ω), h(x) > 1 for all x ∈ Ω

4

and for each h ∈ C+(Ω) we define

h+ := maxΩ

h(x) and h− := minΩh(x).

We denote by M(Ω) the set of real measurable functions defined on Ω.For each p ∈ C+(Ω), we define the generalized Lebesgue space by

Lp(x)(Ω) :=u ∈M(Ω);

∫Ω|u(x)|p(x)dx <∞

.

We consider Lp(x)(Ω) equipped with the Luxemburg norm

|u|p(x) := inf

µ > 0;∫

Ω

∣∣∣∣∣u(x)

µ

∣∣∣∣∣p(x)

dx ≤ 1

.We denote by Lp

′(x)(Ω) the conjugate space of Lp(x)(Ω), where

1

p(x)+

1

p ′(x)= 1, for all x ∈ Ω.

The proof of the following propositions may be found in Bonder and Silva [3], Bon-der, Saintier and Silva [[4], [5]], Fan, Shen and Zhao [10], Fan and Zhang [11] and Fanand Zhao [12].

Proposition 2.1 (Holder Inequality) If u ∈ Lp(x)(Ω) and v ∈ Lp ′(x)(Ω), then

∣∣∣∣∫Ωuvdx

∣∣∣∣ ≤(

1

p−+

1

p ′−

)|u|p(x)|v|p ′(x).

Proposition 2.2 Set ρ(u) :=∫

Ω|u(x)|p(x)dx. For all u ∈ Lp(x)(Ω), we have:

1. For u 6= 0, |u|p(x) = λ ⇔ ρ(uλ) = 1;

2. |u|p(x) < 1 (= 1;> 1) ⇔ ρ(u) < 1 (= 1;> 1);

3. If |u|p(x) > 1, then |u|p−

p(x) ≤ ρ(u) ≤ |u|p+

p(x);

4. If |u|p(x) < 1, then |u|p+

p(x) ≤ ρ(u) ≤ |u|p−

p(x);

5. limk→+∞

|uk|p(x) = 0 ⇔ limk→+∞

ρ(uk) = 0;

6. limk→+∞

|uk|p(x) = +∞ ⇔ limk→+∞

ρ(uk) = +∞.

5

The generalized Lebesgue - Sobolev space W 1,p(x)(Ω) is defined by

W 1,p(x)(Ω) :=u ∈ Lp(x)(Ω); |∇u| ∈ Lp(x)(Ω)

with the norm

‖u‖ := |u|p(x) + |∇u|p(x).

Denoting ρ1,p(x) :=∫

Ω(|u|p(x) + |∇u|p(x))dx ∀u ∈ W 1,p(x)(Ω), we have the following

proposition:

Proposition 2.3 For all u, uj ∈ W 1,p(x)(Ω), we have:

1. ‖u‖ < 1 (= 1;> 1) ⇔ ρ1,p(x)(u) < 1 (= 1;> 1);

2. If ‖u‖ > 1, then ‖u‖p− ≤ ρ1,p(x)(u) ≤ ‖u‖p+;

3. If ‖u‖ < 1, then ‖u‖p+ ≤ ρ1,p(x)(u) ≤ ‖u‖p−;

4. limj→+∞

‖uj‖ = 0 ⇔ limj→+∞

ρ1,p(x)(uj) = 0;

5. limj→+∞

‖uj‖ = +∞ ⇔ limj→+∞

ρ1,p(x)(uj) = +∞.

Proposition 2.4 Suppose that Ω is a bounded smooth domain in IRN and p ∈ C(Ω)with p(x) < N for all x ∈ Ω. If p1 ∈ C(Ω) and 1 ≤ p1(x) ≤ p∗(x) (1 ≤ p1(x) < p∗(x))for x ∈ Ω, then there is a continuous (compact) embedding W 1,p(x)(Ω) → Lp1(x)(Ω),

where p∗(x) =Np(x)

N − p(x).

The Lebesgue spaces on ∂Ω are defined as

Lq(x)(∂Ω) := u| u : ∂Ω→ IR is measurable and∫∂Ω|u|q(x)dS <∞,

and the corresponding (Luxemburg) norm is given by

‖u‖Lq(x)(∂Ω) := ‖u‖q(x),∂Ω := infλ > 0 :∫∂Ω

∣∣∣∣∣u(x)

λ

∣∣∣∣∣q(x)

dS ≤ 1.

Proposition 2.5 Suppose that Ω is a bounded smooth domain in IRN and p, q ∈ C(Ω)with p(x) < N for all x ∈ Ω. Then there is a continuous and compact embedding

W 1,p(x)(Ω) → Lq(x)(∂Ω), where q(x) < p∗(x) =(N − 1)p(x)

N − p(x).

6

Proposition 2.6 (Fan and Zhang [11]) Let Lp(x) : W 1,p(x)(Ω) → (W 1,p(x)(Ω))′ besuch that

Lp(x)(u)(v) =∫

Ω|∇u|p(x)−2∇u∇vdx, ∀ u, v ∈ W 1,p(x)(Ω),

then

(i) Lp(x) : W 1,p(x)(Ω) → (W 1,p(x)(Ω))′ is a continuous, bounded and strictly monotoneoperator;

(ii) Lp(x) is a mapping of type S+, i.e., if un u in W 1,p(x)(Ω) and lim sup(Lp(x)(un)−Lp(x)(u), un − u) ≤ 0, then un → u in W 1,p(x)(Ω);

(iii) Lp(x) : W 1,p(x)(Ω)→ (W 1,p(x)(Ω))′ is a homeomorphism.

Proposition 2.7 (Bonder Saintier and Silva [4])Let (uj)j∈IN ⊂ W 1,p(x)(Ω) be asequence such that uj u weakly in W 1,p(x)(Ω). Then there exists a countable set Ipositive numbers (µ)i∈I and (ν)i∈I and points xi ⊂ A := x ∈ ∂Ω : q(x) = p∗(x)such that

|uj|q(x)dS ν = |u|q(x)dS +∑i∈I

νiδxi , weakly − ∗ in the sense of measures. (2.5)

|∇uj|p(x)dx µ ≥ |∇u|p(x)dx+∑i∈I

µiδxi ,weakly − ∗ in the sense of measures. (2.6)

T xiν1

q(xi)

i ≤ µ1

p(xi)

i , (2.7)

where T xi = supε>0

T (p(.), q(.),Ωε,i,Γε,i) is the localized Sobolev trace constant where

Ωε,i = Ω ∩Bε(xi) and Γε,i = ∂Bε(xi) ∩ Ω.

Proposition 2.8 (Bonder and Silva [3]) Let q(x) and p(x) be two continuous func-tions such that

1 < infx∈Ω

p(x) ≤ supx∈Ω

p(x) < N and 1 ≤ q(x) ≤ p∗(x) in Ω.

Let (uj) be a weakly convergent sequence in W 1,p(x)(Ω) with weak limit u and suchthat:|∇uj|p(x) µ weakly-* in the sense of measures.|uj|q(x) ν weakly-* in the sense of measures.

In addition we assume that A := x ∈ Ω : q(x) = p∗(x) is nonempty. Then, for somecountable index set I, we have:

ν = |u|q(x) +∑i∈I

νiδxi , νi > 0

7

µ ≥ |∇u|p(x) +∑i∈I

µiδi, µi > 0

Sqν1/p∗(xi)i ≤ µ

1/p(xi)i , ∀i ∈ I.

Where xii∈I ⊂ A and Sq is the best constant in the Gagliardo-Nirenberg-Sobolevinequality for variable exponents, namely

Sq = Sq(Ω) = infφ∈C∞0 (Ω)

‖|∇φ|‖Lp(x)(Ω)

‖φ‖Lq(x)(Ω)

.

Definition 2.1 We say that a sequence (uj) ⊂ W 1,p(x)(Ω) is a Palais-Smale sequencefor the functional Jλ,γ if

Jλ,γ(uj)→ c∗ and J ′λ,γ(uj)→ 0 in (W 1,p(x)(Ω))′, (2.8)

wherec∗ = inf

h∈Csupt∈[0,1]

Jλ,γ(h(t)) > 0 (2.9)

and

C = h : [0, 1]→ W 1,p(x)(Ω) : h continuous and h(0) = 0, Jλ,γ(h(1)) < 0.

The number c∗ is called the energy level c∗.When (2.8) implies the existence of a subsequence of (uj), still denoted by (uj),

which converges in W 1,p(x)(Ω), we say that Jλ,γ satisfies the Palais-Smale condition.

3 Proof of Theorem 1.1

Proof. (i) The proof follows from several lemmas.

Lemma 3.1 If (uj) ⊂ W 1,p(x)(Ω) is a Palais-Smale sequence, with energy level c, then(uj) is bounded in W 1,p(x)(Ω).

Proof. Since (uj) is a Palais-Smale sequence with energy level c, we have Jλ,γ(uj)→ cand J ′λ,γ(uj)→ 0. Then taking θ such that

(η + 1)(p+)η+1

(p−)η< θ <

(A1

A2

)r+1 (β−)r+1(r + 1)

(β+)r(3.10)

C + ‖uj‖ ≥(Jλ,γ(uj)−

1

θJ ′λ,γ(uj)uj

)≥

a

(∫Ω

1

p(x)

(|∇uj|p(x) + |uj|p(x)

)dx

)+

b

η + 1

(∫Ω

1

p(x)

(|∇uj|p(x) + |uj|p(x)

)dx

)η+1

8

− λ

r + 1

[∫ΩF (x, uj)dx

]r+1

− γ∫∂Ω

1

q(x)|u|q(x)dS −a

θ

∫Ω|∇uj|p(x)−2∇uj∇ujdx

−aθ

∫Ω|uj|p(x)−2ujujdx+

λ

θ

[∫ΩF (x, uj)

]r ∫Ωf(x, uj)uj +

γ

θ

∫∂Ω|uj|q(x)−2ujujdS

− bθ

(∫Ω

1

p(x)

(|∇uj|p(x) + |uj|p(x)

)dx

)η (∫Ω

(|∇uj|p(x)−2∇uj∇uj + |uj|p(x)−2ujuj

)dx)

Thus,C + ‖uj‖ ≥a

p+

(∫Ω

(|∇uj|p(x) + |uj|p(x)

)dx)

+b

(η + 1)(p+)η+1

(∫Ω

(|∇uj|p(x) + |uj|p(x)

)dx)η+1

− λAr+12

(r + 1)(β−)r+1

[∫Ω|uj|βdx

]r+1

− γ

q−

∫∂Ω|u|q(x)dS − a

θ

∫Ω|∇uj|p(x)dx

−aθ

∫Ω|uj|p(x)dx+

λAr+11

θ(β+)r

[∫Ω|u|β(x)

]r+1

+γ

θ

∫∂Ω|uj|q(x)dS

− b

θ(p−)η

(∫Ω

(|∇uj|p(x) + |uj|p(x)

)dx)η+1

,

and

C + ‖uj‖ ≥(a

p+− a

θ

)(∫Ω

(|∇uj|p(x) + |uj|p(x)

)dx)

+

(b

(η + 1)(p+)η+1− b

θ(p−)η

)(∫Ω

(|∇uj|p(x) + |uj|p(x)

)dx)η+1

+

(λAr+1

1

θ(β+)r− λAr+1

2

(r + 1)(β−)r+1

) [∫Ω|u|β(x)

]r+1

+

(γ

θ− γ

q−

)∫∂Ω|uj|q(x)dS.

Now suppose that (uj) is unbounded in W 1,p(x)(Ω). Thus, passing to a subsequenceif necessary, we get ‖uj‖ > 1 and we obtain

C + ‖uj‖ ≥(a

p+− a

θ

)‖u‖p− +

(b

(η + 1)(p+)η+1− b

θ(p−)η

)‖u‖(η+1)p−

+λ

(Ar+1

1

θ

1

(β+)r− Ar+1

2

r + 1

1

(β−)r+1

)‖u‖β±(r+1),

which is a contradiction because β±(r + 1) > p− > 1. Hence (uj) is bounded inW 1,p(x)(Ω).

Lemma 3.2 Let (uj) ⊂ W 1,p(x)(Ω) be a Palais-Smale sequence, with energy level c and

t0 = limj→∞

(∫Ω

1

p(x)(|∇uj|p(x) + |uj|p(x))dx

). If

c <

(1

θ− 1

q−

)infi∈I

(γ1−1/p(xi)a1/p(xi)T xi

) p(xi)p∗(xi)p∗(xi)−p(xi) ,

where a = t1 with 0 < t1 < btη0, then index set I = ∅ and uj → u strongly in Lq(x)(∂Ω).

9

Proof. Assume that uj u weakly in W 1,p(x)(Ω). By Proposition 2.7 and Lemma 3.1,we have

|uj|q(x)dS ν = |u|q(x)dS +∑i∈I

νiδxi , weakly − ∗ in the sense of measures.

|∇uj|p(x)dx µ ≥ |∇u|p(x)dx+∑i∈I

µiδxi ,weakly − ∗ in the sense of measures.

T xiν1

q(xi)

i ≤ µ1

p(xi)

i , ∀i ∈ I.

If I = ∅ then uj → u strongly in Lq(x)(∂Ω). Suppose that I 6= ∅. Let xi be asingular point of the measures µ and ν. We consider φ ∈ C∞0 (IRN), such that 0 ≤φ(x) ≤ 1, φ(0) = 0 and support in the unit ball of IRN . Consider the functions

φi,ε(x) = φ(x− xiε

)for all x ∈ IRN and ε > 0.

As J ′λ,γ(uj)→ 0 in(W 1,p(x)Ω

)′we obtain that

lim J ′λ,γ(uj)(φi,εuj) = 0.

J ′λ,γ(u)φi,εuj =

(a+ b

(∫Ω

1


)η)×∫

Ω

(|∇uj|p(x)−2∇uj∇φi,εuj + |uj|p(x)−2ujφi,εuj

)dx

−γ∫∂Ω|uj|q(x)−2ujφi,εujdS − λ

[∫ΩF (x, uj)dx

]r ∫Ωf(x, uj)(φi,εuj)dx→ 0,

i.e.,

J ′λ,γ(u)φi,εuj =

(a+ b

(∫Ω

1


)η)×∫

Ω

(|∇uj|p(x)−2∇uj∇φi,εuj + |∇uj|p(x)∇φi,ε + |uj|p(x)φi,ε

)dx

−γ∫∂Ω|uj|q(x)φi,εdS − λ

[∫ΩF (x, uj)dx

]r ∫Ωf(x, uj)(φi,εuj)dx→ 0.

When j →∞ we get

0 = lim

([a+ b

(∫Ω

1


)η] ∫Ω

(|∇uj|p(x)−2∇uj∇φi,εuj)dx[a+ b

(∫Ω

1


)η] ∫Ω|u|p(x)φi,εdx

)+

(a+ btη0)∫

Ωφi,εdµ− γ

∫∂Ωφi,εdν − λ

[∫ΩF (x, u)dx

]r ∫Ωf(x, u)(φi,εu)dx.

10

We may show that,

limε→0

(a+ btη0)∫

Ω(|∇u|p(x)−2∇u∇φi,εu)dx→ 0, see Shang-Wang [18].

On the other hand,

limε→0

(a+ btη0)∫

Ωφi,εdµ = (a+ btη0)µφ(0), γ lim

ε→0

∫∂Ωφi,εdν = γνφ(0)

and

λ[∫

ΩF (x, u)dx

]r ∫Ωf(x, u)(φi,εu)dx→ 0, (a+bt0)

∫Ω|u|p(x)φi,εdx→ 0 as ε→ 0.

Then,(a+ btη0)µiφ(0) = γνiφ(0) implies that γ−1aµi ≤ νi. By T xiν

1/p∗(xi)i ≤ µ

1/p(xi)i we ob-

tain γ−1a(T xi

)p(xi)νp(xi)/p∗(xi)i ≤ γ−1aµi ≤ νi. Thus γ−1a

(T xi

)p(xi) ≤ ν1−p(xi)/p∗(xi)i =

ν(p∗(xi)−p(xi))/p∗(xi)i and γ−1/p(xi)a1/p(xi)T xi ≤ ν

(p∗(xi)−p(xi))/p(xi)p∗(xi)i . Therefore

νi ≥(γ−1/p(xi)a1/p(xi)T xi

) p(xi)p∗(xi)p∗(xi)−p(xi) .

On the other hand, using θ satisfying (3.10)

c = lim Jλ,γ(uj) = lim(Jλ,γ(uj)−

1

θJ ′λ,γ(uj)uj

).

Thus,

c ≥ lim

(γ

θ− γ

q−

)∫∂Ω|uj|q(x)dS

c ≥(γ

θ− γ

q−

)(∫∂Ω|u|q(x)dS +

∫∂Ω

∑i∈I

νiδxidν

)≥(γ

θ− γ

q−

)νi

c ≥(

1

θ− 1

q−

)infi∈I



Therefore, the index set I is empty if, c <

(1

θ− 1

q−

)infi∈I



Lemma 3.3 Let (uj) ⊂ W 1,p(x)(Ω) be a Palais-Smale sequence with energy level c.

If c <

(1

θ− γ

q−

)infi∈I


) p(xi)p∗(xi)p∗(xi)−p(xi) , there exist u ∈ W 1,p(x)(Ω) and a

subsequence, still denoted by (uj), such that uj → u in W 1,p(x)(Ω).

Proof. FromJ ′λ,γ(uj)→ 0,

we have

11

J ′λ,γ(uj)φuj =

(a+ b

(∫Ω

1


)η)×∫

Ω

(|∇uj|p(x)−2∇uj∇(uj − u) + |uj|p(x)−2uj(uj − u)

)dx

−γ∫∂Ω|uj|q(x)−2uj(uj − u)dS

−λ[∫

ΩF (x, uj)dx

]r ∫Ωf(x, uj)(uj − u)dx→ 0,

Note that there exists nonnegative constants c1, c2, c3 and c4 such that

c1 ≤ a+ b

(∫Ω

1


)η≤ c2

and

c3 ≤[∫

ΩF (x, uj)dx

]r≤ c4.

But using Holder inequality, we have∣∣∣∣∫Ω|uj|p(x)−2uj(uj − u)dx

∣∣∣∣ ≤ ∫Ω|uj|p(x)−1|uj − u|dx ≤ C1|u|p(x)/p(x)−1|uj − u|p(x),

and∣∣∣∣∫Ωf(x, uj)(uj − u)dx

∣∣∣∣ ≤ A2

∫Ω|uj|β(x)−1|uj − u|dx ≤ C2|u|β(x)/β(x)−1|uj − u|β(x),

where C1 and C2 are positive constants. Thus∣∣∣∣∫Ω|uj|p(x)−2uj(uj − u)dx

∣∣∣∣→ 0

and ∣∣∣∣∫Ωf(x, uj)(uj − u)dx

∣∣∣∣→ 0.

By Lemma 3.2, uj → u in Lq(x)(∂Ω) and using Holder inequality we obtain∣∣∣∣∫∂Ω|uj|q(x)−2uj(uj − u)

∣∣∣∣ dx→ 0.

If we take

Lp(x)(uj)(uj − u) =∫

Ω|∇uj|p(x)−2∇uj∇(uj − u)dx,

we obtain Lp(x)(uj)(uj − u)→ 0. We also have Lp(x)(u)(uj − u)→ 0. So

(Lp(x)(uj)− Lp(x)(u), uj − u)→ 0.

From Proposition 2.6 we have uj → u in W 1,p(x)(Ω).

12

Lemma 3.4

(i) For all λ > 0, there are α > 0, ρ > 0, such that Jλ,γ(u) ≥ α, ‖u‖ = ρ.

(ii) There exists an element w0 ∈ W 1,p(x)(Ω) with ‖w0‖ > ρ and Jλ,γ(w0) < α.

Proof. (i) We have,

Jλ,γ(u) ≥ a

(∫Ω

1

p(x)(|∇u|p(x) + |u|p(x))dx

)

+b

η + 1

(∫Ω

1

p(x)(|∇u|p(x) + |u|p(x))dx

)η+1

− λ

r + 1

[∫Ω

A2

β(x)uβ(x)dx

]r+1

− γ

q−

∫∂Ω|u|q(x)dS.

So,

Jλ,γ(u) ≥ a

p+

∫Ω

(|∇u|p(x) + |u|p(x))dx

+b

(η + 1)(p+)η+1

(∫Ω

(|∇u|p(x) + |u|p(x))dx)η+1

− λ

r + 1

(A2

β−

)r+1 [∫Ωuβ(x)dx

]r+1

− γ

q−

∫∂Ω|u|q(x)dS.

If ‖u‖ < 1 is small enough, from Proposition 2.3 we obtain∫

Ω(|∇u|p(x) + |u|p(x)) ≥

‖u‖p+ and from the immersions W 1,p(x)(Ω) → Lβ(x)(Ω) and W 1,p(x)(Ω) → Lq(x)(∂Ω)

(continuous), we get∫

Ω|u|β(x)dx ≤Mβ−

1 ‖u‖β−

and∫∂Ω|u|q(x)dx ≤M q−

2 ‖u‖q−

.

Hence

Jλ,γ(u) ≥ a

p+‖u‖p+ +

b

(η + 1)(p+)η+1‖u‖p+(η+1)

− λ

r + 1

(A2

β−

)r+1

Mβ−(r+1)1 ‖u‖β−(r+1) − γ

q−M q−

2 ‖u‖q−.

Using the assumption (η + 1)p+ < β−(r + 1) < q−,

Jλ,γ(u) ≥ a

p+‖u‖(η+1)p+ +

b

(η + 1)(p+)η+1‖u‖p+(η+1)

− λ

r + 1

(A2

β−

)r+1

Mβ−(r+1)1 ‖u‖β−(r+1) − γ

β−(r + 1)M q−

2 ‖u‖β−(r+1),

can be written as

Jλ,γ(u) ≥(a

p++

b

(η + 1)(p+)η+1

)‖u‖p+(η+1)

−

λ

r + 1

(A2

β−

)r+1

Mβ−(r+1)1 +

γ

β−(r + 1)M q−

2

‖u‖β−(r+1).

Taking ρ = ‖u‖,

13

Jλ,γ(u) ≥ ρ(η+1)p+

( a

p++

b

(η + 1)(p+)η+1

)−

λ

r + 1

(A2

β−

)r+1

Mβ−(r+1)1

+γ

β−(r + 1)M q−

2

)ρβ−(r+1)−(η+1)p+

],

and the result follows.

(ii) Take 0 < w ∈ W 1,p(x)(Ω). For t > 1,

Jλ,γ(tw) ≤ a

p−tp

+(∫

Ω(|∇w|p(x) + |w|p(x))dx

)+

btp+(η+1)

(η + 1)(p−)η+1

(∫Ω

(|∇w|p(x) + |w|p(x))dx)η+1

+λ

r + 1

(A2

β−

)r+1

tβ−(r+1)

(∫Ω|w|β(x)dx

)r+1

− γ

q+tq−∫∂Ω|w|q(x)dS.

Then we havelimt→∞

Jλ,γ(tw) = −∞,

and the proof is over.

By Lemma 3.4, we may use the Mountain Pass Theorem [19], which guarantees theexistence of a sequence (uj) ⊂ W 1,p(x)(Ω) such that

Jλ,γ(uj)→ c and J ′λ,γ(uj)→ 0 in (W 1,p(x)(Ω))′,

where a candidate for critical value is

c = infh∈C

supt∈[0,1]

Jλ,γ(h(t))

and C = h : [0, 1]→ W 1,p(x)(Ω) : h continuous and h(0) = 0, h(1) = w0.For 0 < t < 1 and fixing w ∈ W 1,p(x)(Ω),

Jλ,γ(tw) ≤ a

p−tp−(∫

Ω(|∇w|p(x) + |w|p(x))dx

)+

btp−(η+1)

(η + 1)(p−)η+1

(∫Ω

(|∇w|p(x) + |w|p(x))dx)η+1

− λ

r + 1

(A1

β+

)tβ

+(r+1)(∫

Ω|w|β(x)

)r+1

− γ

q+tq

+∫∂Ω|w|q(x)dS.

Jλ,γ(tw) ≤ a

p−tp−(∫

Ω(|∇w|p(x) + |w|p(x))dx

)+

btp−(η+1)

(η + 1)(p−)η+1

(∫Ω

(|∇w|p(x) + |w|p(x))dx)η+1

− λ

r + 1

(A1

β+

)tβ

+(r+1)(∫

Ω|w|β(x)

)r+1

.

14

Setting g(t) =

(aa

p−+

baη+1

(η + 1)(p−)η+1

)tp− − λ

r + 1

(A1

β+

)btβ−(r+1),

where a =∫

Ω(|∇w|p(x) + |w|p(x))dx and b =

∫Ω|w|β(x)dx, we obtain sup Jλ(tw) ≤ g(t).

Note that g(t) has a critical point of maximum

tλ =

[β+((η + 1)(p−)η)aa+ b(a)η+1)

λA1(η + 1)(p−)η bβ−

] 1β+(r+1)−p−

,

and that tλ → 0 when λ→∞. By the continuity of Jλ,γ

limλ→∞

(supt≥0

Jλ,γ(tw)

)= 0.

Then exists λ0 such that ∀λ ≥ λ0

supt≥0

Jλ,γ(tw) <

(1

θ− γ

q−

)infi∈I


) p(xi)p∗(xi)p∗(xi)−p(xi)

This completes the proof of part (i).

Proof.(ii) Like the item (i).

Proof. (iii) The proof follows from several lemmas.

Lemma 3.5 Let (uj) ⊂ W 1,p(x)(Ω) be a Palais-Smale sequence for Jλ,γ then (uj) isbounded.

Proof. Let (uj) be a Palais-Smale sequence with energy level c, i.e., Jλ,γ(uj)→ c andJ ′λ,γ(uj)→ 0. Then taking θ such that

m1p+

m0

< θ <(A1

A2

)κ+1 (β−)κ+1(κ+ 1)

(β+)κ, (3.11)

we have

C + ‖uj‖ ≥(Jλ,γ(uj)−

1

θJ ′λ,γ(uj)uj

).

So

C + ‖uj‖ ≥(m0

p+− m1

θ

)(∫Ω

(|∇uj|p(x) + |uj|p(x)

)dx)

+

(λ

θ− λ

q−

)∫Ω|uj|q(x)dx(

γAκ+11

θ(β+)κ− γAκ+1

2

(κ+ 1)(β−)κ+1

) [∫∂Ω|u|β(x)dS

]κ+1

.

15

Now suppose that (uj) is unbounded in W 1,p(x)(Ω). Thus, passing to a subsequenceif necessary, we get ‖uj‖ > 1 and we obtain

C + ‖uj‖ ≥(

1

p+− 1

θ

)‖uj‖p

−,

which is a contradiction because p− > 1. Hence (uj) is bounded in W 1,p(x)(Ω).

Lemma 3.6 Let (uj) ⊂ W 1,p(x)(Ω) be a Palais-Smale sequence, with energy level c. If

c <

(1

θ− 1

q−A

)λ(m0S)N , where m0 = min(λ−1m0)1/p+ , (λ−1m0)1/p−, then index set

I = ∅ and uj → u strongly in Lq(x)(Ω).

Proof. Assume that uj u weakly in W 1,p(x)(Ω). By Proposition 2.8 and Lemma 3.5,we have

|uj|q(x) ν = |u|q(x) +∑i∈I

νiδxi , νi > 0

|∇uj|p(x) µ ≥ |∇u|p(x) +∑i∈I

µiδxi , µi > 0

Sν1/p∗(xi)i ≤ µ

1/p(xi)i , ∀i ∈ I.

If I = ∅ then uj → u strongly in Lq(x)(Ω). Suppose that I 6= ∅. Let xi be a singularpoint of the measures µ and ν. We consider φ ∈ C∞0 (IRN), such that 0 ≤ φ(x) ≤ 1,φ(0) = 0 and and support in the unit ball of IRN . Consider the functions φi,ε(x) =

φ(x− xiε

)for all x ∈ IRN and ε > 0.

As J ′λ,γ(uj)→ 0 in(W 1,p(x)Ω

)′we obtain that

lim J ′λ,γ(uj)(φi,εuj) = 0.

J ′λ,γ(u)(φi,εuj) = M

(∫Ω

1


)×∫

Ω

(|∇uj|p(x)−2∇uj∇φi,εuj + |uj|p(x)−2ujφi,εuj

)dx

−λ∫

Ω|uj|q(x)−2ujφi,εujdx− γ

[∫∂ΩG(x, uj)dS

]κ ∫∂Ωg(x, uj)(φi,εuj)dS → 0.

When j →∞ we get

0 = lim

(M

(∫Ω

1


)∫Ω

(|∇uj|p(x)−2∇uj∇φi,εuj)dx

M

(∫Ω

1


)∫Ω|uj|p(x)φi,εdx

)+

16

M(t0)∫

Ωφi,εdµ− λ

∫Ωφi,εdν − γ

[∫∂ΩG(x, u)dS

]κ ∫∂Ωg(x, u)(φi,εu)dS,

where t0 = limj→∞

(∫Ω

1


).

We may show that,

limε→0

M(t0)∫

Ω(|∇u|p(x)−2∇u∇φi,εu)dx→ 0, see Shang-Wang [18].

On the other hand,

limε→0

M(t0)∫

Ωφi,εdµ = M(t0)µφ(0), λ lim

ε→0

∫Ωφi,εdν = νφ(0)

and

γ[∫∂ΩG(x, u)dS

]κ ∫∂Ωg(x, u)(φi,εu)dS → 0, M(t0)

∫Ω|u|p(x)φi,εdx→ 0 quando ε→

0.Then,M(t0)µiφ(0) = λνiφ(0) implies that λ−1m0µi ≤ νi. By Sν

1/p∗(xi)i ≤ µ

1/p(xi)i we get

(m0S)N ≤ νi,

where m0 = min(λ−1m0)1/p+ , (λ−1m0)1/p−.On the other hand, using θ satisfying (3.11) we have

c = lim(Jλ(uj)−

1

θJ ′λ,γ(uj)uj

).

Note that,

c ≥ lim∫

Ω

(λ

θ− λ

q(x)

)|uj|q(x)dx.

Considering Aδ =⋃x∈A

(Bδ(x) ∩ Ω) = x ∈ Ω : dist(x,A) < δ,

c ≥(λ

θ− λ

q−Aδ

)(∫Aδ|u|q(x)dx+

∑i∈I

νi

)≥(λ

θ− λ

q−Aδ

)νi.

Therefore,

c ≥(λ

θ− λ

q−A

)(m0S)N .

Therefore, the index set I is empty if,

c <

(1

θ− 1

q−A

)λ(m0S)N .

17

Lemma 3.7 Let (uj) ⊂ W 1,p(x)(Ω) be a Palais-Smale sequence with energy level c. If

c <

(1

θ− 1

q−A

)λ(m0S)N , there exist u ∈ W 1,p(x)(Ω) and a subsequence, still denoted

by (uj), such that uj → u in W 1,p(x)(Ω).

Proof. FromJ ′λ,γ(uj)→ 0,

we have

J ′λ,γ(uj)(uj − u) = M

(∫Ω

1


)×∫

Ω

(|∇uj|p(x)−2∇uj∇(uj − u) + |uj|p(x)−2uj(uj − u)

)dx

−λ∫

Ω|uj|q(x)−2uj(uj − u)dx

−γ[∫∂ΩG(x, uj)dS

]κ ∫∂Ωg(x, uj)(uj − u)dS → 0.

Note that there exists nonnegative constants c1 and c2 such that

c1 ≤[∫∂ΩG(x, uj)dS

]r≤ c2.

Using Holder inequality, one has

∣∣∣∣∫Ω|uj|p(x)−2uj(uj − u)dx

∣∣∣∣ ≤ ∫Ω|uj|p(x)−1|uj − u|dx ≤ C1||u|p(x)−1|p(x)/p(x)−1|uj − u|p(x),

and∣∣∣∣∫∂Ωg(x, uj)(uj − u)dS

∣∣∣∣ ≤ A2

∫∂Ω|uj|β(x)−1|uj − u|dS

≤ C2||u|β(x)−1|β(x)/β(x)−1|uj − u|β(x),

where C1 and C2 are positive constants. Thus∣∣∣∣∫Ω|uj|p(x)−2uj(uj − u)dx

∣∣∣∣→ 0

and ∣∣∣∣∫∂Ωg(x, uj)(uj − u)dS

∣∣∣∣→ 0.

By Lemma 3.6, uj → u in Lq(x)(Ω) and using Holder inequality we obtain∣∣∣∣∫Ω|uj|q(x)−2uj(uj − u)dx

∣∣∣∣→ 0.

If we take

18

Lp(x)(uj)(uj − u) =∫

Ω|∇uj|p(x)−2∇uj∇(uj − u),

we obtain Lp(x)(uj)(uj − u)→ 0. We also have Lp(x)(u)(uj − u)→ 0. So

(Lp(x)(uj)− Lp(x)(u), uj − u)→ 0.

From Proposition 2.6 we have uj → u in W 1,p(x)(Ω).

Lemma 3.8

(i) For all λ > 0, there are α > 0, ρ > 0, such that Jλ,γ(u) ≥ α, ‖u‖ = ρ.

(ii) There exists an element w0 ∈ W 1,p(x)(Ω) with ‖w0‖ > ρ and Jλ,γ(w0) < α.

Proof. (i) We have,

Jλ,γ(u) ≥ m0

(∫Ω

1

p(x)(|∇u|p(x) + |u|p(x))dx

)− γ

κ+ 1

[∫∂Ω

A2

β(x)uβ(x)dS

]κ+1

− λ

q−

∫Ω|u|q(x)dx.

So,

Jλ,γ(u) ≥ m0

p+

∫Ω

(|∇u|p(x) + |u|p(x))dx− γ

κ+ 1

(A2

β−

)κ+1 [∫∂Ωuβ(x)dS

]κ+1

− λ

q−

∫Ω|u|q(x)dx.

If ‖u‖ < 1 is small enough, from Proposition 2.3 we obtain∫

Ω(|∇u|p(x) + |u|p(x)) ≥

‖u‖p+ and from the immersions W 1,p(x)(Ω) → Lβ(x)(Ω) and W 1,p(x)(Ω) → Lq(x)(∂Ω)

(continuous), we get∫

Ω|u|β(x)dx ≤Mβ−

1 ‖u‖β−

and∫∂Ω|u|q(x)dx ≤M q−

2 ‖u‖q−

.

Hence

Jλ,γ(u) ≥ m0

p+‖u‖p+ − γ

κ+ 1

(A2

β−

)κ+1

Mβ−(κ+1)1 ‖u‖β−(κ+1) − λ

q−M q−

2 ‖u‖q−.

Using the assumption p+ < β−(κ+ 1) < q−,

Jλ,γ(u) ≥ m0

p+‖u‖p+ − γ

κ+ 1

(A2

β−

)κ+1

Mβ−(κ+1)1 ‖u‖β−(κ+1)

− λ

β−(κ+ 1)M q−

2 ‖u‖β−(κ+1),

can be written as

Jλ,γ(u) ≥ m0

p+‖u‖p+ −

γ

κ+ 1

(A2

β−

)κ+1

Mβ−(κ+1)1 +

λ

β−(κ+ 1)M q−

2

‖u‖β−(κ+1).

Taking ρ = ‖u‖,

19

Jλ,γ(u) ≥ ρp+

(m0

p+

)−

γ

r + 1

(A2

β−

)κ+1

Mβ−(κ+1)1 +

λ

β−(κ+ 1)M q−

2

ρβ−(κ+1)−p+ ,

and the result follows.

(ii) Take 0 < w ∈ W 1,p(x)(Ω). For t > 1,

Jλ,γ(tw) ≤ m1

p−tp

+(∫

Ω(|∇w|p(x) + |w|p(x))dx

)− λ

q+tq−∫

Ω|w|q(x)dx

+γ

κ+ 1

(A2

β−

)κ+1

tβ−(κ+1)

(∫∂Ω|w|β(x)dS

)κ+1

.

Then we havelimt→∞

Jλ,γ(tw) = −∞,

and the proof is over.

By Lemma 3.4, we may use the Mountain Pass Theorem [19], which guarantees theexistence of a sequence (uj) ⊂ W 1,p(x)(Ω) such that

Jλ,γ(uj)→ c and J ′λ,γ(uj)→ 0 in (W 1,p(x)(Ω))′,

where a candidate for critical value is

c = infh∈C

supt∈[0,1]

Jλ,γ(h(t))

and C = h : [0, 1]→ W 1,p(x)(Ω) : h continuous and h(0) = 0, h(1) = w0.For 0 < t < 1 and fixing w ∈ W 1,p(x)(Ω),

Jλ,γ(tw) ≤ m1

p−tp−(∫

Ω(|∇w|p(x) + |w|p(x))dx

)− λ

q+tq

+∫

Ω|w|q(x)dx

− γ

κ+ 1

(A1

β+

)κ+1

tβ+(κ+1)

(∫∂Ω|w|β(x)dS

)κ+1

.

Jλ,γ(tw) ≤ m1

p−tp−(∫

Ω(|∇w|p(x) + |w|p(x))dx

)− γ

κ+ 1

(A1

β+

)κ+1

tβ+(κ+1)

(∫∂Ω|w|β(x)dS

)κ+1

.

Settingg(t) =m1a

p−tp− − γ

κ+ 1

(A1

β+

)κ+1

btβ+(κ+1),

where a =∫

Ω(|∇w|p(x) + |w|p(x))dx and b =

∫Ω|w|β(x)dx, we obtain sup Jλ,γ(tw) ≤ g(t).

Note that g(t) has a critical point of maximum

tγ =

[m1(β+)κa

γAκ1 b

] 1β+(κ+1)−p−)

,

20

and that tγ → 0 when γ →∞. By the continuity of Jλ,γ

limγ→∞

(supt≥0

Jλ,γ(tw)

)= 0.

Then exists γ1 such that ∀γ ≥ γ1

supt≥0

Jλ,γ(tw) <

(1

θ− 1

q−A

)λ(m0S)N .

This completes the proof of part (iii).

Proof.(iv) Similar to the proof of the item (iii).

Remark 3.1 In a forthcoming article we will present results concerning multiplicity ofsolutions for the problem (1.1).

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