By a direct variational approach and the theory of the variable exponent Sobolev spaces, under growth conditions on the reaction terms, we establish the existence and multiplicity of solutions

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

EXISTENCE OF SOLUTIONS FOR NONLOCAL ELLIPTIC SYSTEMS WITH NONSTANDARD GROWTH CONDITIONS

GUOWEI DAI

Abstract. This article concerns the existence and multiplicity of solutions for ap(x)-Kirchhoff-type systems with Dirichlet boundary condition. By a direct variational approach and the theory of the variable exponent Sobolev spaces, under growth conditions on the reaction terms, we establish the existence and multiplicity of solutions.

1. Introduction

In this article, we study the following nonlocal elliptic systems of gradient type with nonstandard growth conditions

−M1

Z

Ω

1

p(x)|∇u|^p(x)dx

div |∇u|^p(x)−2∇u

= ∂F

∂u(x, u, v) in Ω,

−M2

Z

Ω

1

q(x)|∇v|^q(x)dx

div |∇v|^q(x)−2∇v

=∂F

∂v(x, u, v) in Ω, u= 0, v= 0 on∂Ω,

(1.1)

where Ω is a bounded domain in R^N with a smooth boundary ∂Ω, p(x), q(x) ∈ C+(Ω) with

1< p⁻:= min

Ω

p(x)≤p⁺ := max

Ω

p(x)<+∞, 1< q⁻:= min

Ω

q(x)≤q⁺:= max

Ω

q(x)<+∞,

M1(t), M2(t) are continuous functions. We confine ourselves to the case where M1 = M2 for simplicity. Notice that the results of this paper remain valid for M1 6= M2 by adding some slight changes in the hypothesis (H4) and (H5). The functionF: Ω×R×R→Ris assumed to be continuous inx∈Ω and of classC¹ inu, v ∈R.

The operator −div(|∇u|^p(x)−2∇u) is called the p(x)-Laplacian, and becomes p-Laplacian whenp(x)≡p(a constant). Thep(x)-Laplacian possesses more com- plicated nonlinearities than thep-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition

2000Mathematics Subject Classification. 35D05, 35J60, 35J70.

Key words and phrases. Variational method; nonlinear elliptic systems; nonlocal condition.

c

2011 Texas State University - San Marcos.

Submitted July 9, 2011. Published October 19, 2011.

Supported by grants 11061030 from the NSFC, and NWNU-LKQN-10-21.

1

has been received considerable attention in recent years. These problems are in- teresting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field [1, 34, 37]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non- Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [5, 6]. Another field of applica- tion of equations with variable exponent growth conditions is image processing [9].

The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [13, 29, 35, 38, 39] for an overview of and references on this subject, and to [2, 20, 21, 22, 23, 24, 25, 26] for the study of thep(x)-Laplacian equations and the corresponding variational problems.

Problem (1.1) is related to the stationary version of a model introduced by Kirchhoff [30]. More precisely, Kirchhoff proposed the model

ρ∂²u

∂t² −ρ0

h + E 2L

Z L

0

|∂u

∂x|²dx∂²u

∂x² = 0, (1.2)

where ρ, ρ₀, h, E, L are constants, which extends the classical D’Alembert’s wave equation, by considering the effects of the changes in the length of the strings during the vibrations. A distinguishing feature of equation (1.2) is that the equation contains a nonlocal coefficient ^ρ_h⁰ +_2L^E RL

0 |^∂u_∂x|²dx which depends on the average

1 2L

RL

0 |^∂u_∂x|²dx, and hence the equation is no longer a pointwise identity. Some early classical studies of Kirchhoff equations were Bernstein [7] and Pohoˇzaev [33]. The equation

− a+b

Z

Ω

|∇u|²dx

∆u=f(x, u) in Ω, u= 0 on∂Ω,

(1.3) is related to the stationary analogue of the equation (1.2). Equation (1.3) received much attention only after Lions [31] proposed an abstract framework to the prob- lem. Some important and interesting results can be found, for example, in [3, 8, 17].

More recently Alves et al. [4] and Ma and Rivera [32] obtained positive solutions of such problems by variational methods. The study of Kirchhoff type equations has already been extended to the case involving the p-Laplacian (for details, see [10, 18, 19])and p(x)-Laplacian (see [12, 15, 17, 27]). In [12], by a direct varia- tional approach, we establish conditions ensuring the existence and multiplicity of solutions for the problem

−MZ

Ω

1

p(x)|∇u|^p(x)dx

div(|∇u|^p(x)−2∇u) =f(x, u) in Ω, u= 0 on∂Ω.

In [28], the author established that existence and multiplicity results for a class of elliptic systems with nonstandard growth conditions.

Motivated by above, we consider the nonlocal elliptic systems (1.1). We establish the existence and multiplicity of solutions for system (1.1). Local elliptic systems with standard growth conditions have been the subject of a sizeable literature. We

refer to the excellent survey article by De Figueiredo [14]. We also refer to [11]

about nonlocal elliptic systems ofp-Kirchhoff-type.

This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Sections 3, we give some existence results of weak solutions of problem (1.1) and their proofs.

2. Preliminaries

To discuss problem (1.1), we need some theory on W₀^1,p(x)(Ω) which is called variable exponent Sobolev space. Firstly we state some basic properties of spaces W₀^1,p(x)(Ω) which will be used later (for details, see [25]). Denote byS(Ω) the set of all measurable real functions defined on Ω. Two functions inS(Ω) are considered as the same element ofS(Ω) when they are equal almost everywhere. Write

C+(Ω) ={h:h∈C(Ω), h(x)>1 for any x∈Ω}, h⁻:= min

Ω

h(x), h⁺:= max

Ω

h(x) for every h∈C+(Ω).

Define

L^p(x)(Ω) ={u∈S(Ω) : Z

Ω

|u(x)|^p(x)dx <+∞forp∈C+(Ω)}

with the norm

|u|_Lp(x)(Ω)=|u|_p(x)= inf{λ >0 : Z

Ω

|u(x)

λ |^p(x)dx≤1}, and

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

with the norm

kuk_W1,p(x)(Ω)=|u|_Lp(x)(Ω)+|∇u|_Lp(x)(Ω). Denote byW₀^1,p(x)(Ω) the closure ofC₀^∞(Ω) inW^1,p(x)(Ω).

Proposition 2.1 ([25]). The spacesL^p(x)(Ω),W^1,p(x)(Ω)andW₀^1,p(x)(Ω)are sep- arable and reflexive Banach spaces.

Proposition 2.2 ([25]). Setρ(u) =R

Ω|u(x)|^p(x)dx. For anyu∈L^p(x)(Ω), then (1) foru6= 0,|u|_p(x)=λif and only ifρ(^u_λ) = 1;

(2) |u|_p(x)<1 (= 1;>1) if and only ifρ(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) lim_k→+∞|uk|_p(x)= 0 if and only iflim_k→+∞ρ(uk) = 0;

(6) lim_k→+∞|uk|_p(x)= +∞ if and only iflim_k→+∞ρ(uk) = +∞.

Proposition 2.3([25]). InW₀^1,p(x)(Ω)the Poincar´e inequality holds; that is, there exists a positive constantC0 such that

|u|_Lp(x)(Ω)≤C0|∇u|_Lp(x)(Ω), ∀u∈W₀^1,p(x)(Ω).

So,|∇u|_Lp(x)(Ω) is a norm equivalent to the norm kuk in the space W₀^1,p(x)(Ω).

We will use the equivalent norm in the following discussion and write kukp =

|∇u|_Lp(x)(Ω)for simplicity.

Proposition 2.4 ([22, 25]). If q ∈ C+(Ω) and q(x) ≤ p^∗(x) (q(x) < p^∗(x)) for x ∈ Ω, then there is a continuous (compact) embedding W₀^1,p(x)(Ω) ,→ L^q(x)(Ω), where

p^∗(x) =

( _{N p(x)}

N−p(x) if p(x)< N, +∞ if p(x)≥N.

Proposition 2.5 ([23, 25]). The conjugate space of L^p(x)(Ω) is L^q(x)(Ω), where

1

q(x)+_p(x)¹ = 1 holds a.e. in Ω. For any u∈L^p(x)(Ω) andv ∈L^q(x)(Ω), we have the following H¨older-type inequality

Z

Ω

uv dx ≤( 1

p⁻ + 1

q⁻)|u|_p(x)|v|_q(x). We write

I(u) = Z

Ω

1

p(x)|∇u|^p(x)dx.

Proposition 2.6 ([23]). The functional I : X → R is convex. The mapping I⁰:X →X^∗ is a strictly monotone, bounded homeomorphism, and is of(S+)type, namely

u_n* uand lim sup

n→+∞

I⁰(u_n)(u_n−u)≤0 impliesu_n→u, whereX =W₀^1,p(x)(Ω),X^∗ is the dual space ofX.

For every (u, v) and (ϕ, ψ) in W :=W₀^1,p(x)(Ω)×W₀^1,q(x)(Ω), let F(u, v) :=

Z

Ω

F(x, u, v)dx.

Then

F⁰(u, v)(ϕ, ψ) =D1F(u, v)(ϕ) +D2F(u, v)(ψ), where

D1F(u, v)(ϕ) = Z

Ω

∂F

∂u(x, u, v)ϕ dx, D2F(u, v)(ψ) =

Z

Ω

∂F

∂v(x, u, v)ψ dx.

The Euler-Lagrange functional associated to (1.1) is given by J(u, v) :=McZ

Ω

1

p(x)|∇u|^p(x)dx +Mc(

Z

Ω

1

q(x)|∇v|^q(x)dx)− F(u, v), where Mc(t) :=Rt

0M(τ)dτ. It is easy to verify that J ∈C¹(W,R) is weakly lower semi-continuous and (u, v)∈W is a weak solution of (1.1) if and only if (u, v) is a critical point ofJ. Moreover, we have

J⁰(u, v)(ϕ, ψ) =D₁J(u, v)(ϕ) +D₂J(u, v)(ψ), (2.1) where

D1J(u, v)(ϕ) =MZ

Ω

1

p(x)|∇u|^p(x)dxZ

Ω

|∇u|^p(x)−2∇u∇ϕ dx−D1F(u, v)(ϕ), D₂J(u, v)(ψ) =MZ

Ω

1

q(x)|∇v|^q(x)dxZ

Ω

|∇v|^q(x)−2∇v∇ψ dx−D₂F(u, v)(ψ).

Let us choose onW the normk · kdefined by

k(u, v)k:= max{kuk_p,kvk_q}.

The dual space of W will be denoted by W^∗ and k · k_∗ will stand for its norm.

Therefore

kJ⁰(u, v)k_∗=kD1J(u, v)k_∗,p+kD2J(u, v)k_∗,q

where k · k_∗,p (respectively k · k_∗,q) is the norm of (W₀^1,p(x)(Ω))^∗ (respectively (W₀^1,q(x)(Ω))^∗).

3. Existence of solutions

In this section we discuss the existence of weak solutions of (1.1). For simplicity, we usec, ci, i = 1,2, . . . to denote the general positive constant (the exact value may change from line to line).

Before stating our results, we introduce some natural growth hypotheses on the right-hand side of (1.1) and the nonlocal coefficient M(t). These hypotheses will ensure the mountain pass geometry and the Palais-Smale condition for the Euler- Lagrange functionalJ.

(H1) For all (x, s, t)∈Ω×R², we assume

|F(x, s, t)| ≤c

1 +|s|^p1(x)+|t|^q1(x)+|s|^α(x)|t|^β(x) ,

where c is a positive constant, (p1(x), q1(x), α(x), β(x)) ∈ (C+(Ω))⁴ such that

p₁(x)< p^∗(x), q₁(x)< q^∗(x), 2α(x)

p^∗(x) +2β(x)

q^∗(x) <1 in Ω, p⁻₁, 2α⁻> p⁺, q⁻₁, 2β⁻> q⁺.

(H2) There exist M > 0,θ1 > _1−µ^p+ , θ2 > _1−µ^q+ such that for all x∈Ω, and all (s, t)∈R²with|s|^θ1+|t|^θ2≥2M, one has

0< F(x, s, t)≤ s θ1

∂F

∂s(x, s, t) + t θ2

∂F

∂t(x, s, t), whereµcomes from (H5) below.

(H3) F(x, s, t) =o(|s|^p++|t|^q+) as (s, t) →(0,0) uniformly with respect to to x∈Ω.

(H4) There existsm₀>0, such thatM(t)≥m₀.

(H5) There exists 0< µ <1 such thatMc(t)≥(1−µ)M(t)t.

As an example, we let M(t) = a+bt : R⁺ → R with a, b are two positive constants. It is clear thatM(t)≥a >0. Takingµ= 1/2, we have

Mc(t) = Z t

0

M(s)ds=at+1 2bt²≥ 1

2(a+bt)t= (1−µ)M(t)t.

So conditions (H4), (H5) are satisfied.

Theorem 3.1. If M satisfies(H4) and

|F(x, s, t)| ≤c₁(1 +|s|^α1+|t|^β1),

whereα1,β1are two constants with1≤α1<min{p⁻, q⁻},1≤β1<min{p⁻, q⁻} then (1.1)has a weak solution.

Proof. From (H4) we haveMc(t)≥m0t. For (un, vn)∈W such thatk(un, vn)k → +∞, we have

J(u_n, v_n)

=Mc( Z

Ω

1

p(x)|∇un|^p(x)dx) +Mc( Z

Ω

1

q(x)|∇vn|^q(x)dx)− Z

Ω

F(x, un, vn)dx

≥m₀ Z

Ω

1

p(x)|∇u_n|^p(x)dx+m₀ Z

Ω

1

q(x)|∇v_n|^q(x)dx

−c₁ Z

Ω

|un|^α1dx−c₁ Z

Ω

|vn|^β1dx−c₁|Ω|

≥ m₀

p⁺kunk^p_p⁻+m₀

q⁺kvnk^q_q⁻−c3kunk^α_p¹−c2kvnk^β_q¹−c1|Ω|,

where |Ω| denotes the measure of Ω. Without loss of generality, we may assume ku_nk_p≥ kv_nk_q. Hence,

J(u_n, v_n)≥m₀

p⁺kunk^p_p⁻−c₃kunk^α_p¹−c₂kunk^β_p¹−c₁|Ω|, (3.1) By the definition of norm on W, we have k(un, vn)k =kunkp →+∞. In view of (3.1) and the assumptions onα1 and β1, we can easily see thatJ(un, vn)→+∞

as n → +∞; i.e., J is a coercive functional. Since J also is weakly lower semi- continuous,J has a minimum point (u, v) inW, and (u, v) is a weak solution pair which may be trivial of (1.1). The proof is completed.

Lemma 3.2. Let(u_n, v_n)be a Palais-Smale sequence for the Euler-Lagrange func- tionalJ. If(H2), (H4), (H5) are satisfied then(u_n, v_n)is bounded.

Proof. Let (un, vn) be a Palais-Smale sequence for the functional J. This means that J(un, vn) is bounded and kJ⁰(un, vn)k∗ → 0 as n → +∞. Then, there is a positive constantc0 such that

c₀≥J(u_n, v_n)

=McZ

Ω

1

p(x)|∇un|^p(x)dx

+McZ

Ω

1

q(x)|∇vn|^q(x)dx

− Z

Ω

F(x, un, vn)dx

≥(1−µ)MZ

Ω

1

p(x)|∇un|^p(x)dxZ

Ω

1

p(x)|∇un|^p(x)dx

− Z

Ω

u_n θ1

∂F

∂u(x, un, vn)dx+ (1−µ)MZ

Ω

1

q(x)|∇vn|^q(x)dx

× Z

Ω

1

q(x)|∇vn|^q(x)dx− Z

Ω

vn

θ₂

∂F

∂v(x, un, vn)dx−c4, wherec4is some positive constant. Then

c₀≥J(u_n, v_n)

≥ 1−µ p⁺ − 1

θ₁ MZ

Ω

1

p(x)|∇un|^p(x)dxZ

Ω

|∇un|^p(x)dx+ 1

θ₁D1J(un, vn)(un) + 1−µ

q⁺ − 1 θ₂

MZ

Ω

1

q(x)|∇v_n|^q(x)dxZ

Ω

|∇v_n|^q(x)dx + 1

θ₂D2J(un, vn)(vn)−c4

≥ 1−µ p⁺ − 1

θ₁ m0

Z

Ω

|∇un|^p(x)dx+ 1−µ q⁺ − 1

θ₂ m0

Z

Ω

|∇vn|^q(x)dx

− 1 θ1

kD1J(u_n, v_n)k∗,pkunkp− 1 θ2

kD2J(u_n, v_n)k∗,qkvnkq−c₄.

Now, suppose that the sequence (un, vn) is not bounded. Without loss of generality, we may assumekunkp≥ kvnkq.

Therefore, fornlarge enough, we have c5≥ 1−µ

p⁺ − 1 θ1

m0kunk^p_p⁻−1 θ1

kD1J(un, vn)k_∗,p+ 1 θ2

kD2J(un, vn)k_∗,q kunkp. But, this cannot hold true sincep⁻>1. Hence,{k(un, vn)k}is bounded.

In the following lemma, we show every bounded Palais-Smale sequence for the functionalJ contains a convergence subsequence.

Lemma 3.3. Let (un, vn) be a bounded Palais-Smale sequence for the Euler-La- grange functionalJ. If(H1), (H4)are satisfied, then(un, vn)contains a convergent subsequence.

Proof. Let (u_n, v_n) be a bounded Palais-Smale sequence for the functionalJ. Then there is a subsequence still denoted by (u_n, v_n) which converges weakly in W. Without loss of generality, we assume that (u_n, v_n)*(u, v), thenJ⁰(u_n, v_n)(u_n− u, vn−v)→0. Thus, we have

J⁰(u_n, v_n)(u_n−u, v_n−v)

=MZ

Ω

1

p(x)|∇un|^p(x)dxZ

Ω

|∇un|^p(x)−2∇un(∇un− ∇u)dx +MZ

Ω

1

q(x)|∇vn|^q(x)dxZ

Ω

|∇vn|^q(x)−2∇vn(∇vn− ∇v)dx

− Z

Ω

∂F

∂u(x, u_n, v_n)(u_n−u)dx− Z

Ω

∂F

∂v(x, u_n, v_n)(v_n−v)dx→0.

On the other hand, letα,e βebe two continuous and positive functions on Ω such that

2α(x) +α(x)e

p^∗(x) +2β(x) +β(x)e

q^∗(x) = 1, ∀x∈Ω.

Using the Young inequality, we obtain

|s|^α(x)|t|^β(x)≤ |s| ^α(x)p

∗(x) 2α(x)+eα(x) +|t|

β(x)q∗(x)

2β(x)+eβ(x) =|s|^p2(x)+|t|^q2(x), where p₂(x) := _2α(x)+^α(x)p∗(x)

α(x)e < p^∗(x), q₂(x) := ^β(x)q∗(x)

2β(x)+eβ(x) < q^∗(x). From (H1), we can obtain that there existp3(x), q3(x)∈C+(Ω) withp3(x)< p^∗(x),q3(x)< q^∗(x) in Ω such that

|F(x, s, t)| ≤c₆ 1 +|s|^p3(x)+|t|^q3(x) . From this inequality, Propositions 2.4 and 2.5, we can easily obtain

Z

Ω

∂F

∂u(x, u_n, v_n)(u_n−u)dx→0

and Z

Ω

∂F

∂v(x, un, vn)(vn−v)dx→0. (3.2)

Therefore, we have MZ

Ω

1

p(x)|∇un|^p(x)dxZ

Ω

|∇un|^p(x)−2∇un(∇un− ∇u)dx→0, MZ

Ω

1

q(x)|∇vn|^q(x)dxZ

Ω

|∇vn|^q(x)−2∇vn(∇vn− ∇v)dx→0.

In view of (H4), we have Z

Ω

|∇u_n|^p(x)−2∇u_n(∇u_n− ∇u)dx→0, Z

Ω

|∇v_n|^q(x)−2∇v_n(∇v_n− ∇v)dx→0.

Using Proposition 2.6, we haveun →uin W₀^1,p(x)(Ω) and vn → v in W₀^1,q(x)(Ω), which implies that (un, vn)→(u, v) inW. This completes the proof.

Theorem 3.4. If hypotheses (H1)–(H5) hold, then (1.1) has at least one weak solution.

Proof. Let us show thatJ satisfies the conditions of Mountain Pass Theorem (see [36, Theorem 2.10]). By Lemmas 3.2 and 3.3,J satisfies Palais-Smale condition in W.

Fork(u, v)k <1, using the Young’s inequality, the fact ^2α(x)_p∗(x)+^2β(x)_q∗(x) <1 in Ω, Propositions 2.2 and 2.4, we obtain

Z

Ω

|u|^α(x)|v|^β(x)dx≤ 1 2

Z

Ω

|u|^2α(x)dx+1 2

Z

Ω

|v|^2β(x)dx≤c7(kuk^2α_p ⁻+kvk^2β_q ⁻).

On the other hand, assuming (H1), W₀^1,p(x)(Ω) ,→ L^p+(Ω), and W₀^1,q(x)(Ω) ,→ L^q+(Ω). Then there existsc8, c9>0 such that

|u|_p+ ≤c8kukp foru∈W₀^1,p(x)(Ω)

|v|_q+≤c9kvkq forv∈W₀^1,q(x)(Ω),

where| · |rdenote the norm onL^r(x)(Ω) withr∈C+(Ω). Letε >0 be small enough such thatεc^p₈⁺ ≤_2p^m+⁰ andεc^q₉⁺≤ _2q^m+⁰. By the assumptions (H1) and (H3), we have

|F(x, s, t)| ≤ε |s|^p++|t|^q+

+c(ε)(|s|^p1(x)+|t|^q1(x)+|s|^α(x)|t|^β(x)) for all (x, s, t)∈Ω×R². In view of (H4) and and the above inequality, fork(u, v)k sufficiently small, noting Proposition 2.2, we have

J(u, v)≥ m0

p⁺ Z

Ω

|∇u|^p(x)dx+m0

q⁺ Z

Ω

|∇v|^q(x)dx−ε Z

Ω

|u|^p+dx−ε Z

Ω

|v|^q+dx

−c(ε) Z

Ω

|u|^p1(x)+|v|^q1(x)+|u|^α(x)|v|^β(x) dx

≥ m0

p⁺kuk^p_p⁺−εc^p₈⁺kuk^p_p⁺+m0

q⁺kvk^q_q⁺−εc^q₉⁺kvk^q_q⁺

−c(ε) kuk^p

−

p1 +kvk^q

−

q1 +c₇kuk^2α_p ⁻+c₇kvk^2β_q ⁻

≥ m₀

2p⁺kuk^p_p⁺+ m₀

2q⁺kvk^q_q⁺−c(ε)

kuk^pp⁻¹ +kvk^qq⁻¹ +c₇kuk^2α_p ⁻+c₇kvk^2β_q ⁻ .

Sincep⁻₁,2α⁻ > p⁺ andq₁⁻,2β⁻ > q⁺, there existr >0,δ >0 such that J(u)≥ δ >0 for everyk(u, v)k=r.

On the other hand, we have known that the assumption (H2) implies the follow- ing assertion: for everyx∈Ω,s, t∈R, the inequality

F(x, s, t)≥c₁₀(|s|^θ1+|t|^θ2−1) (3.3) holds; see [28]. Whent > t0, from (H5) we can easily obtain that

Mc(t)≤ Mc(t0) t^1/(1−µ)₀

t^1/(1−µ):=c11t^1/(1−µ),

wheret₀ is an arbitrarily positive constant. For (eu,ev)∈W\ {(0,0)}andt >1, we have

J(teu, tv) =e McZ

Ω

1

p(x)|t∇eu|^p(x)dx

+McZ

Ω

1

q(x)|t∇ev|^q(x)dx

− Z

Ω

F(x, tu, te ev)dx

≤c12( Z

Ω

|t∇u|e^p(x)dx)^1/(1−µ)−c10t^θ1 Z

Ω

|eu|^θ1dx +c13

Z

Ω

|t∇ev|^q(x)dx1/(1−µ)

−c10t^θ2 Z

Ω

|ev|^θ2dx−c14

≤c12t^p

+ 1−µ

Z

Ω

|∇u|e^p(x)dx1/(1−µ)

−c10t^θ1 Z

Ω

|u|e^θ1dx +c13t ^q

+ 1−µ

Z

Ω

|∇ev|^q(x)dx1/(1−µ)

−c10t^θ2 Z

Ω

|ev|^θ2dx−c14

→ −∞, as t→+∞,

due toθ1> _1−µ^p+ andθ2> _1−µ^q+ . SinceJ(0,0) = 0, considering Lemmas 3.2 and 3.3, we see that J satisfies the conditions of Mountain Pass Theorem. SoJ admits at

least one nontrivial critical point.

Next we will prove under some symmetry condition on the functionF that (1.1) possesses infinitely many nontrivial weak solutions.

Theorem 3.5. Assume(H1), (H2), (H4), (H5), and thatF(x, u, v)is even inu,v.

Then (1.1)has a sequence of solutions{(±u_k,±v_k)}^∞_k=1 such that J(±u_k,±v_k)→ +∞ask→+∞.

Because W₀^1,p(x) and W₀^1,q(x) are a reflexive and separable Banach space, then W andW^∗ are too. There exist{ej} ⊂W and{e^∗_j} ⊂W^∗ such that

W = span{ej :j= 1,2, . . .}, W^∗= span{e^∗_j :j= 1,2, . . .}, and

hei, e^∗_ji=

(1, i=j, 0, i6=j,

where h·,·idenotes the duality product between W andW^∗. For convenience, we write Xj = span{ej}, Yk = ⊕^k_j=1Xj, Zk = ⊕^∞_j=kXj. We will use the following

“Fountain theorem” to prove Theorem 3.5.

Lemma 3.6 ([36]). Assume

(A1) X is a Banach space,I∈C¹(X,R)is an even functional.

(A2) For eachk= 1,2, . . ., there existρk> rk>0 such that (A2) inf_{u∈Zk,kuk=rk}I(u)→+∞ask→+∞.

(A3) max_{u∈Yk,kuk=ρk}I(u)≤0.

(A4) I satisfies Palais-Smale condition for everyc >0.

ThenI has a sequence of critical values tending to+∞.

For everya >1,u, v∈L^a(Ω), we define

|(u, v)|a := max{|u|a,|v|a}.

Set

a:= max

x∈Ω

{2α(x),2β(x), p₁(x), q₁(x)}>min{p⁻, q⁻}, b:= min

x∈Ω

{2α(x),2β(x), p1(x), q1(x)}>0.

Then we have the following result.

Lemma 3.7 ([28]). Denote

βk = sup{|(u, v)|a:k(u, v)k= 1,(u, v)∈Zk}.

Thenlim_k→+∞β_k = 0.

Proof of Theorem 3.5. According to the assumptions onF, Lemmas 3.2 and 3.3, J is an even functional and satisfies Palais-Smale condition. We will prove that if k is large enough, then there existρk > rk >0 such that (A2) and (A3) holding.

Thus, the conclusion can be obtained from Fountain theorem.

(A2): For any (uk, vk)∈Zk,kukkp ≥1,kvkkq ≥1 andk(uk, vk)k=rk (rk will be specified below), we have

J(uk, vk)

=McZ

Ω

1

p(x)|∇uk|^p(x)dx

+McZ

Ω

1

q(x)|∇vk|^q(x)dx

− Z

Ω

F(x, uk, vk)dx

≥m0

Z

Ω

1

p(x)|∇uk|^p(x)dx+m0

Z

Ω

1

q(x)|∇vk|^q(x)dx− Z

Ω

F(x, uk, vk)dx

≥ m0

p⁺ Z

Ω

|∇u_k|^p(x)dx+m0

q⁺ Z

Ω

|∇v_k|^q(x)dx

−c Z

Ω

1 +|uk|^p1(x)+|vk|^q1(x)+|uk|^α(x)|vk|^β(x) dx

≥ m₀

p⁺kukk^p_p⁻+m₀

q⁺kvkk^q_q⁻−c|uk|^p_p^1(ξk1)

1(x) −c|vk|^q_q^1(ξ2k)

1(x)

−c15|uk|^2α(η_2α(x)^1k)−c15|vk|^2β(η_2β(x)^k2)−c|Ω|, whereξ^k₁, ξ₂^k, η₁^k, η₂^k∈Ω. Therefore, J(uk, vk)

≥ m0

max{p⁺, q⁺}k(uk, vk)k^{min{p−,q−}}−c|uk|^pa^1(ξk1)−c|vk|^qa^1(ξk2)

−c|uk|^2α(ηa ^k1)−c|vk|^2β(ηa ^k2)−c|Ω|

≥ m₀

max{p⁺, q⁺}k(uk, v_k)k^{min{p−,q−}}−c(β_kk(uk, v_k)k)^p1(ξk1)−c(β_kk(uk, v_k)k)^q1(ξ2k)

−c(βkk(uk, vk)k)^2α(ηk1)−c(βkk(uk, vk)k)^2β(ηk2)−c|Ω|

≥ m0

max{p⁺, q⁺}k(u_k, v_k)k^{min{p−,q−}}−c₁₆β^b_kk(u_k, v_k)k^a−c|Ω|, wherea,b are defined above. At this stage, we fixrk as follows:

rk := m0

2c16max{p⁺, q⁺}β_k^b

1/(a−min{p⁻,q⁻})

→+∞ ask→+∞.

Consequently, ifk(uk, vk)k=rk then J(uk, vk)≥ m0

2 max{p⁺, q⁺}k(uk, vk)k^{min{p−,q−}}−c|Ω| →+∞ ask→+∞.

(A3): From (H2), we haveF(x, u, v)≥c10(|u|^θ1+|v|^θ2−1) for everyx∈Ω and u, v ∈R. Therefore, for any (u, v)∈ Yk with k(u, v)k = 1 and 1< ρk =tk with tk →+∞, we have

J(t_ku, t_kv)

=McZ

Ω

1

p(x)|tk∇u|^p(x)dx

+McZ

Ω

1

q(x)|tk∇v|^q(x)dx

− Z

Ω

F(x, tku, tkv)dx.

≤c₁₇Z

Ω

|t_k∇u|^p(x)dx1/(1−µ)

+c₁₈Z

Ω

|t_k∇v|^q(x)dx1/(1−µ)

−c10t^θ_k¹ Z

Ω

|u|^θ1dx−c10t^θ_k² Z

Ω

|v|^θ2dx+c19,

≤c17t

p+ 1−µ

k

Z

Ω

|∇u|^p(x)dx1/(1−µ)

−c10t^θ_k¹ Z

Ω

|u|^θ1dx +c₁₈t

q+ 1−µ

k

Z

Ω

|∇v|^q(x)dx^1/(1−µ)

−c₁₀t^θ_k² Z

Ω

|v|^θ2dx+c₁₉.

Byθ₁> _1−µ^p+ ,θ₂>_1−µ^q+ and dimY_k =k, it is easy to see thatJ(t_ku, t_kv)→ −∞as k(t_ku, t_kv)k →+∞for (u, v)∈Y_k.

The proof of Theorem 3.5 is completed by the Fountain theorem.

Acknowledgments. The author wishes to express his gratitude to the anonymous referee for reading the original manuscript carefully and making several corrections and remarks.

References

[1] E. Acerbi and G. Mingione; Regularity results for stationary electro-rheologicaluids, Arch.

Ration. Mech. Anal., 164 (2002), 213–259.

[2] E. Acerbi, G. Mingione;Gradient estimate for thep(x)-Laplacean system, J. Reine Angew.

Math., 584 (2005), 117–148.

[3] A. Arosio, S. Panizzi;On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc.

348 (1996) 305–330.

[4] C. O. Alves, F. J. S. A. Corrˆea, T. F. Ma;Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005) 85–93.

[5] S. N. Antontsev, S. I. Shmarev;A Model Porous Medium Equation with Variable Exponent of Nonlinearity: Existence, Uniqueness and Localization Properties of Solutions, Nonlinear Anal. 60 (2005), 515–545.

[6] S. N. Antontsev , J. F. Rodrigues;On Stationary Thermo-rheological Viscous Flows, Ann.

Univ. Ferrara, Sez. 7, Sci. Mat. 52 (2006), 19–36.

[7] S. Bernstein;Sur une classe d’´equations fonctionnelles aux d´eriv´ees partielles, Bull. Acad.

Sci. URSS. Ser. Math. [Izv. Akad. Nauk SSSR] 4 (1940) 17–26.

[8] M. M. Cavalcanti, V. N. Cavalcanti, J.A. Soriano;Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations 6 (2001) 701–730.

[9] Y. Chen, S. Levine, M. Rao;Variable exponent, linear growth functionals in image restora- tion, SIAM J. Appl.Math. 66 (4) (2006), 1383–1406.

[10] F. J. S. A. Corrˆea, G. M. Figueiredo;On a elliptic equation ofp-kirchhoff type via variational methods, Bull. Austral. Math. Soc. Vol. 74 (2006) 263–277.

[11] F. J. S. A. Corrˆea, R.G. Nascimento;On a nonlocal elliptic system ofp-Kirchhoff-type under Neumann boundary condition, Mathematical and Computer Modelling 49 (2009) 598–604.

[12] G. Dai, R. Hao; Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal.

Appl. 359 (2009) 275–284.

[13] L. Diening, P. H¨ast¨o, A. Nekvinda;Open problems in variable exponent Lebesgue and Sobolev spaces, in: P. Dr´abek, J. R´akosn´ık, FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp.38–58.

[14] G. Dai, D. Liu;Infinitely many positive solutions for ap(x)-Kirchhoff-type equation, J. Math.

Anal. Appl. 359 (2009) 704–710.

[15] G. Dai, J. Wei;Infinitely many non-negative solutions for ap(x)-Kirchhoff-type problem with Dirichlet boundary condition, Nonlinear Analysis 73 (2010) 3420–3430.

[16] P. D’Ancona, S. Spagnolo;Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992) 247–262.

[17] D. G. De Figueiredo;Semilinear elliptic systems: a survey of superlinear problems, Resenhas 2 (1996) 373–391.

[18] M. Dreher; The Kirchhoff equation for the p-Laplacian, Rend. Semin. Mat. Univ. Politec.

Torino 64 (2006) 217–238.

[19] M. Dreher;The wave equation for thep-Laplacian, Hokkaido Math. J. 36 (2007) 21–52.

[20] X. L. Fan;On the sub-supersolution methods forp(x)-Laplacian equations, J. Math. Anal.

Appl. 330 (2007), 665–682.

[21] X. L. Fan and X. Y. Han;Existence and multiplicity of solutions forp(x)-Laplacian equations inR^N, Nonlinear Anal. 59 (2004), 173–188.

[22] X. L. Fan, J. S. Shen, D. Zhao;Sobolev embedding theorems for spacesW^k,p(x)(Ω), J. Math.

Anal. Appl. 262 (2001), 749–760.

[23] X. L. Fan, Q. H. Zhang;Existence of solutions for p(x)-Laplacian Dirichlet problems, Non- linear Anal. 52 (2003), 1843–1852.

[24] X. L. Fan, Q. H. Zhang and D. Zhao;Eigenvalues of p(x)-Laplacian Dirichlet problem, J.

Math. Anal. Appl. 302 (2005), 306–317.

[25] X.L. Fan, D. Zhao; On the Spaces L^p(x) and W^m,p(x), J. Math. Anal. Appl. 263 (2001), 424–446.

[26] X. L. Fan, Y. Z. Zhao, Q. H. Zhang;A strong maximum principle forp(x)-Laplace equations, Chinese J. Contemp. Math. 24 (3) (2003) 277–282.

[27] X. L. Fan;On nonlocalp(x)-Laplacian Dirichlet problems, Nonlinear Anal. 72 (2010) 3314–

3323.

[28] A. El Hamidi;Existence results to elliptic systems with nonstandard growth conditions, J.

Math. Anal. Appl. 300 (2004) 30–42.

[29] P. Harjulehto, P. H¨ast¨o; An overview of variable exponent Lebesgue and Sobolev spaces, in Future Trends in Geometric Function Theory (D. Herron (ed.), RNC Work-shop), Jyv¨askyl¨a, 2003, 85–93.

[30] G. Kirchhoff;Mechanik, Teubner, Leipzig, 1883.

[31] J. L. Lions; On some equations in boundary value problems of mathematical physics, in:

Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North- Holland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346.

[32] T. F. Ma, J. E. Mu˜noz Rivera;Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003) 243–248.

[33] S. I. Pohoˇzaev;A certain class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) 96 (1975) 152–166, 168.

[34] M. R ˙u˘zi˘cka;Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.

[35] S. Samko;On a progress in the theory of Lebesgue spaces with variable exponent Maximal and singular operators, Integral Transforms Spec. Funct. 16 (2005), 461–482.

[36] M. Willem;Minimax Theorems, Birkh¨auser, Boston, 1996.

[37] V. V. Zhikov; Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 9 (1987), 33–66.

[38] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik;Homogenization of Differential Operators and Integral Functionals, Translated from the Russian by G.A. Yosifian, Springer-Verlag, Berlin, 1994.

[39] V. V. Zhikov;On some variational problems, Russian J. Math. Phys. 5 (1997), 105–116.

Guowei Dai

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China E-mail address:[email protected]