Multiple Solutions for a Class of Differential Inclusion System Involving the (𝑝(𝑥),𝑞(𝑥))-Laplacian

Abstract

We consider a differential inclusion system involving the (𝑝(𝑥),𝑞(𝑥))-Laplacian with Dirichlet boundary condition on a bounded domain and obtain two nontrivial solutions under appropriate hypotheses. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.

1. Introduction

In recent years, the study of differential equations and variational problems with 𝑝(𝑥)-growth conditions has been a new and interesting topic, which arises from nonlinear electrorheological fluids (see [1]) and elastic mechanics (see [2]). The study on variable exponent problems attracts more and more interest in recent years, and many results have been obtained on this kind of problems, for example [3–11].

Elliptic systems with standard growth conditions have been the subject of a sizeable literature. We refer to the excellent survey article of de Figueiredo [12].

In [11], the author obtained the existence and multiplicity of solutions for the following problem:−div|∇𝑢|^{𝑝(𝑥)−2}∇𝑢=𝐹_𝑢(𝑥,𝑢,𝑣),in||Ω,−div∇𝑣^{𝑞(𝑥)−2}∇𝑣=𝐹_𝑣(𝑥,𝑢,𝑣),inΩ,𝑢=𝑣=0,on𝜕Ω, (𝑃₁) where Ω⊂ℝ^𝑁 is a bounded domain with a smooth boundary 𝜕Ω, 𝑁≥2, (𝑝,𝑞)∈𝐶(Ω)², 𝑝(𝑥)>1, 𝑞(𝑥)>1, for every 𝑥∈Ω. The function 𝐹 is assumed to be continuous in 𝑥∈Ω and of class 𝐶¹ in 𝑢,𝑣∈ℝ. More precisely, the author was able to prove that, under suitable conditions, the system might have at least one solution or have infinite number of solutions.

Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, now a question arises: whether there exist solutions for system (𝑃₁) in the case where there is no continuously differentiable hypothesis required on the potential function 𝐹 with respect to 𝑡(𝑠). See, for example, 𝐹(𝑥,𝑡,𝑠) is locally Lipschitz with respect to 𝑡(𝑠). That is the main problem which we want to solve in the present paper.

To this end, we mainly discuss the existence and multiplicity of solutions for the following nonlinear differential inclusion system involving the (𝑝(𝑥),𝑞(𝑥))-Laplacian:−div|∇𝑢|^{𝑝(𝑥)−2}∇𝑢∈𝜆𝜕_𝑢𝐹(𝑥,𝑢,𝑣),in||Ω,−div∇𝑣^{𝑞(𝑥)−2}∇𝑣∈𝜆𝜕_𝑣𝐹(𝑥,𝑢,𝑣),inΩ,𝑢=𝑣=0,on𝜕Ω, (𝑃) where Ω⊂ℝ^𝑁 is a bounded domain with 𝐶¹-boundary 𝜕Ω, 𝜆>0 is the parameter, 𝑝,𝑞∈𝐶(Ω), 1<𝑝⁻≤𝑝⁺<+∞, 1<𝑞⁻≤𝑞⁺<+∞, 𝐹∶Ω×ℝ×ℝ→ℝ is a function such that 𝐹(⋅,𝑡,𝑠) is measurable in Ω for all (𝑡,𝑠)∈ℝ×ℝ, and 𝐹(𝑥,𝑡,𝑠) is locally Lipschitz with respect to 𝑡(𝑠) (in general it can be nonsmooth), 𝜕_𝑡𝐹(𝑥,𝑡,𝑠)(𝜕_𝑠𝐹(𝑥,𝑡,𝑠)) is the subdifferential with respect to the 𝑡(𝑠)-variable in the sense of Clarke [13].

We emphasize that the operator −div(|∇𝑢|^{𝑝(𝑥)−2}∇𝑢) is said to be 𝑝(𝑥)-Laplacian, which becomes 𝑝-Laplacian when 𝑝(𝑥)≡𝑝 (a constant). The 𝑝(𝑥)-Laplacian possesses more complicated nonlinearities than the 𝑝-Laplacian, for example, it is inhomogeneous and, in general, it does not have the first eigenvalue. In other words, the infimum of the eigenvalues of 𝑝(𝑥)-Laplacian equals 0 (see [14]).

Specially, if 𝐹(𝑥,⋅,𝑣),𝐹(𝑥,𝑢,⋅)∈𝐶¹(ℝ) for a.a. 𝑥∈Ω, and 𝑝(𝑥)=𝑝, 𝑞(𝑥)=𝑞, then the problem (𝑃) becomes the following problem:−div|∇𝑢|^𝑝−2∇𝑢=𝜆𝐹_𝑢(𝑥,𝑢,𝑣),in||Ω,−div∇𝑣^𝑞−2∇𝑣=𝜆𝐹_𝑣(𝑥,𝑢,𝑣),inΩ,𝑢=𝑣=0,on𝜕Ω. (𝑃₂) There have been a large number of papers that study the existence of the solutions to (𝑃₂). For instance, when 𝑝>𝑁,𝑞>𝑁, Li and Tang [15] ensured the existence of three solutions to this problem. In [16], Kristály studied the multiplicity of solutions of the quasilinear elliptic systems (𝑃₁), where Ω is a strip-like domain and 𝜆>0 is the parameter. Under some growth conditions on 𝐹, the author guaranteed the existence of an open interval Λ⊂[0,+∞), such that for each 𝜆∈Λ problem (𝑃₂) has at least two distinct nontrivial solutions; when 𝑝 and 𝑞 are real numbers larger than 1, 𝜆=1, Boccardo and Guedes de Figueiredo [17] obtained the existence of solutions of the system (𝑃₂).

But up to now, to the best of our knowledge, no paper discussing the solutions of problem (𝑃) with nonsmooth potential via nonsmooth critical point theory can be found in the existing literature. In order to fill in this gap, we study problem (𝑃) from a more extensive viewpoint. More precisely, we would study the existence of at least two nontrivial solutions for the problem (𝑃) as the parameter 𝜆>𝜆₀ for some constant 𝜆₀.

This paper is divided into three sections: in the second section we introduce some necessary knowledge on the nonsmooth analysis, basic properties of the generalized Lebesgue-space 𝐿^𝑝(𝑥)(Ω) and the generalized Lebesgue-Sobolev space 𝑊^1,𝑝(𝑥)(Ω). In the third section, we give the assumptions on the nonsmooth potential 𝐹(𝑥,𝑡,𝑠) and prove the multiplicity results for problem (𝑃).

2. Preliminary

2.1. Variable Exponent Sobolev Space

In order to discuss problem (𝑃), we need some theories on 𝑊₀^{𝑘,𝑝(𝑥)}(Ω) which we call variable exponent Sobolev space. Firstly we review some facts on variable exponent spaces 𝐿^𝑝(𝑥)(Ω) and 𝑊^{𝑘,𝑝(𝑥)}(Ω). For the details see [4, 18–20].

Firstly, we need to give some notations, which we shall use through this paper:𝐶₊Ω=𝑝∈𝐶Ω∶𝑝(𝑥)>1forany𝑥∈Ω,𝑝⁻=lim_𝑥∈Ω𝑝(𝑥),𝑝⁺=max_𝑥∈Ω𝑝(𝑥)forany𝑝∈𝐶₊Ω. (2.1) Obviously, 1<𝑝⁻≤𝑝⁺<+∞.

Denote by 𝒰(Ω) the set of all measurable real functions defined on Ω. Two functions in 𝒰(Ω) are considered to be one element of 𝒰(Ω), when they are equal almost everywhere.

For 𝑝∈𝐶₊(Ω), define𝐿^𝑝(𝑥)(Ω)=𝑢∈𝒰(Ω)∶_Ω|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥<+∞, (2.2) and with the norm |𝑢|_{𝐿𝑝(𝑥)(Ω)}=|𝑢|_𝑝(𝑥)=inf{𝜆>0∶∫_Ω|𝑢(𝑥)/𝜆|^𝑝(𝑥)𝑑𝑥≤1},𝑊^1,𝑝(𝑥)(Ω)={𝑢∈𝐿^𝑝(𝑥)𝐿(Ω)∶|∇𝑢|∈_𝑝(𝑥)(Ω)}, with the norm ‖𝑢‖=‖𝑢‖_{𝑊1,𝑝(𝑥)(Ω)}=|𝑢|_𝑝(𝑥)+|∇𝑢|_𝑝(𝑥).

Denote 𝑊₀^1,𝑝(𝑥)(Ω) as the closure of 𝐶^∞₀(Ω) in 𝑊^1,𝑝(𝑥)(Ω).

Hereafter, let 𝑝^∗(𝑥)=𝑁𝑝(𝑥)(𝑥)(𝑥)(𝑥)𝑁−𝑝,𝑝<𝑁,+∞,𝑝≥𝑁. (2.3)

Lemma 2.1 (see [19]). (1) The spaces 𝐿^𝑝(𝑥)(Ω), 𝑊^1,𝑝(𝑥)(Ω), and 𝑊₀^1,𝑝(𝑥)(Ω) are separable and reflexive Banach spaces. Moreover, 𝐿^𝑝(𝑥)(Ω) is uniform convex.
(2) Poincare inequality in 𝑊₀^1,𝑝(𝑥)(Ω) holds; that is, there exists a positive constant 𝐶 such that |𝑢|_{𝐿𝑝(𝑥)(Ω)}≤𝐶|∇𝑢|_{𝐿𝑝(𝑥)(Ω)},∀𝑢∈𝑊₀^1,𝑝(𝑥)(Ω). (2.4)
(3) If 𝑞∈𝐶₊(Ω) and 𝑞(𝑥)<𝑝^∗(𝑥) for any 𝑥∈Ω, then the embedding from 𝑊^1,𝑝(𝑥)(Ω) to 𝐿^𝑞(𝑥)(Ω) is compact and continuous.

By (2) of Lemma 2.1, we know that |∇𝑢|_𝑝(𝑥) and ‖𝑢‖ are equivalent norms on 𝑊₀^1,𝑝(𝑥)(Ω). We will use |∇𝑢|_𝑝(𝑥) to replace ‖𝑢‖ in the following discussions.

Lemma 2.2 (see [4]). The conjugate space of 𝐿^𝑝(𝑥)(Ω) is 𝐿^𝑞(𝑥)(Ω), where 1/𝑝(𝑥)+1/𝑞(𝑥)=1. For any 𝑢∈𝐿^𝑝(𝑥)(Ω) and 𝑣∈𝐿^𝑞(𝑥)(Ω), one has _Ω||1𝑢𝑣𝑑𝑥≤𝑝⁻+1𝑞⁻|𝑢|_{𝐿𝑝(𝑥)(Ω)}|𝑣|_{𝐿𝑞(𝑥)(Ω)}. (2.5)

Lemma 2.3 (see [4]). Set 𝜌(𝑢)=∫_Ω|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥. For 𝑢,𝑢_𝑘∈𝐿^𝑝(𝑥)(Ω), one has(1)for 𝑢≠0,|𝑢|_𝑝(𝑥)=𝜆⇔𝜌(𝑢/𝜆)=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)lim_𝑘→+∞|𝑢_𝑘|_𝑝(𝑥)=0⇔Lim_𝑘→+∞𝜌(𝑢_𝑘)=0; (6)|𝑢_𝑘|_𝑝(𝑥)→+∞⇔𝜌(𝑢_𝑘)→+∞.

In this paper, the space 𝑊₀^1,𝑝(𝑥)(Ω)×𝑊₀^1,𝑞(𝑥)(Ω) will be endowed with the following equivalent norm:‖(𝑢,𝑣)‖=‖𝑢‖+‖𝑣‖, (2.6) where‖𝑢‖=inf𝜆>0∶_Ω∇𝑢𝜆^𝑝(𝑥)𝑑𝑥≤1,‖𝑣‖=inf𝜆>0∶_Ω∇𝑣𝜆^𝑞(𝑥)𝑑𝑥≤1. (2.7)

Similar to Lemma 2.3, we have the following.

Lemma 2.4. Set 𝜌(𝑢)=∫_Ω|∇𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥. For 𝑢,𝑢_𝑘∈𝑊^1,𝑝(𝑥)(Ω), one has(1)for 𝑢≠0,‖𝑢‖=𝜆⇔𝜌(𝑢/𝜆)=1;(2)‖𝑢‖<1(=1;>1)⇔𝜌(𝑢)<1(=1;>1); (3)if ‖𝑢‖>1, then ‖𝑢‖^𝑝−≤𝜌(𝑢)≤‖𝑢‖^𝑝+;(4)if ‖𝑢‖<1, then ‖𝑢‖^𝑝+≤𝜌(𝑢)≤‖𝑢‖^𝑝−;(5)Lim_𝑘→+∞‖𝑢_𝑘‖=0⇔Lim_𝑘→+∞𝜌(𝑢_𝑘)=0; (6)‖𝑢_𝑘‖→+∞⇔𝜌(𝑢_𝑘)→+∞.
Consider the following function:𝐽(𝑢)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥,𝑢∈𝑊₀^1,𝑝(𝑥)(Ω). (2.8) We know that (see [21]) 𝐽∈𝐶¹(𝑊₀^1,𝑝(𝑥)(Ω),𝑅) and 𝑝(𝑥)-Laplacian operator −Δ_𝑝(𝑥)𝑢=−div(|∇𝑢|^{𝑝(𝑥)−2}∇𝑢) is the derivative operator of 𝐽 in the weak sense. We denote ℒ=𝐽′∶𝑊₀^1,𝑝(𝑥)(Ω)→(𝑊₀^1,𝑝(𝑥)(Ω))^∗, then ⟨ℒ(𝑢),𝑣⟩=∫_Ω(|∇𝑢(𝑥)|^{𝑝(𝑥)−2}∇𝑢⋅∇𝑣𝑑𝑥,forall𝑢,𝑣∈𝑊₀^1,𝑝(𝑥)(Ω).

Lemma 2.5 (see [19]). Set 𝑋=𝑊₀^1,𝑝(𝑥)(Ω), ℒ is as above, then(1)ℒ∶𝑋→𝑋^∗ is a continuous, bounded and strictly monotone operator;(2)ℒ is a mapping of type (𝑆₊), if 𝑢_𝑛⇀𝑢(weak) in 𝑋 andlimsup_𝑛→∞⟨ℒ(𝑢_𝑛),𝑢_𝑛−𝑢⟩≤0, then 𝑢_𝑛→𝑢 in 𝑋;(3)ℒ∶𝑋→𝑋^∗ is a homeomorphism.

2.2. Generalized Gradient

Let 𝑋 be a Banach space, 𝑋^∗ its topological dual space and we denote ⟨⋅,⋅⟩ as the duality bracket for pair (𝑋^∗,𝑋). A function 𝜑∶𝑋↦ℝ is said to be locally Lipschitz, if for every 𝑥∈𝑋 we can find a neighbourhood 𝑈 of 𝑥 and a constant 𝑘>0(depending on 𝑈), such that |𝜑(𝑦)−𝜑(𝑧)|≤𝑘‖𝑦−𝑧‖,forall𝑦,𝑧∈𝑈.

The generalized directional derivative of 𝜑 at the point 𝑢∈𝑋 in the direction ℎ∈𝑋 is𝜑⁰(𝑢;ℎ)=limsup_{𝑢′→𝑢;𝜆↓0}𝜑𝑢^′+𝜆ℎ−𝜑𝑢^′𝜆. (2.9)

The generalized subdifferential of 𝜑 at the point 𝑢∈𝑋 is defined by the𝜕𝜑(𝑢)=𝑢^∗∈𝑋^∗;𝑢^∗,ℎ≤𝜑⁰(𝑢;ℎ),∀ℎ∈𝑋, (2.10) which is a nonempty, convex and 𝑤^∗-compact set of 𝑋. We say that 𝑢∈𝑋 is a critical point of 𝜑, if 0∈𝜕𝜑(𝑢). For further details, we refer the reader to [12].

Finally we have the following Weierstrass Theorem and Mountain Pass Theorem.

Theorem 2.6. If 𝑋 is a reflexive Banach space and 𝜑∶𝑋→ℝ satisfies(1)𝜑 is weak lower semicontinuous, that is, 𝑥_𝑛⇀𝑥₀weaklyin𝑋⟶𝜑𝑥₀≤liminf_𝑛→+∞𝜑𝑥_𝑛; (2.11)(2)𝜑 is coercive, that is, lim_{‖𝑥‖→∞}𝜑(𝑥)=+∞,then there exists 𝑥^∗∈𝑋 such that 𝜑(𝑥^∗)=min_𝑥∈𝑋𝜑(𝑥).

Theorem 2.7 (see [22]). Let 𝜑∶𝑋→ℝ be locally Lipschitz function and 𝑥₀,𝑥₁∈𝑋. If there exists a bounded open neighbourhood 𝑈 of 𝑥₀, such that 𝑥₁∈𝑋⧵𝑈, max{𝜑(𝑥₀),𝜑(𝑥₁)}<inf_𝜕𝑈𝜑 and 𝜑 satisfies the nonsmooth C-condition at level 𝑐, where 𝑐=inf_𝛾∈𝒯max_𝑡∈[0,1]𝜑(𝛾(𝑡)),𝒯={𝛾∈𝐶([0,1];𝑋)∶𝛾(0)=𝑥₀,𝛾(1)=𝑥₁}, then 𝑐 is a critical value of 𝜑 and 𝑐≥inf_𝜕𝑈𝜑.

3. Existence Results

For each (𝑢,𝑣)∈𝑊₀^1,𝑝(𝑥)(Ω)×𝑊₀^1,𝑞(𝑥)(Ω), defineΦ(𝑢,𝑣)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥+_Ω1𝑝(𝑥)|∇𝑣|^𝑞(𝑥)Ψ()𝑑𝑥,𝑢,𝑣=_Ω𝐹(𝑥,𝑢,𝑣)𝑑𝑥. (3.1)

By a solution of (2), we mean function (𝑢,𝑣)∈𝑊₀^1,𝑝(𝑥)(Ω)×𝑊₀^1,𝑞(𝑥)(Ω) to which there corresponds mapping Ω∋𝑥→(𝑤₁,𝑤₂) with 𝑤₁(𝑥)∈𝜕_𝑢𝐹(𝑥,𝑢,𝑣), 𝑤₂(𝑥)∈𝜕_𝑣𝐹(𝑥,𝑢,𝑣) for almost every 𝑥∈Ω having the property that for every (𝜉,𝜂)∈𝑊₀^1,𝑝(𝑥)(Ω)×𝑊₀^1,𝑞(𝑥)(Ω), the function 𝑥→(𝑤₁(𝑥)𝜉(𝑥),𝑤₂(𝑥)𝜂(𝑥))∈𝐿¹(Ω)×𝐿¹(Ω) and_Ω|∇𝑢|^{𝑝(𝑥)−2}∇𝑢∇𝜉𝑑𝑥+_Ω|∇𝑣|^{𝑞(𝑥)−2}∇𝑣∇𝜂𝑑𝑥=𝜆_Ω𝑤₁𝜉𝑑𝑥+𝜆_Ω𝑤₂𝜂𝑑𝑥. (3.2)

Our hypotheses on nonsmooth potential 𝐹(𝑥,𝑡,𝑠) is as follows.

H(F): 𝐹∶Ω×ℝ×ℝ→ℝ is a function such that 𝐹(𝑥,0,0)=0 a.e. on Ω and satisfies the following facts:(1)for all 𝑡∈ℝ, 𝑠∈ℝ, 𝑥↦𝐹(𝑥,𝑡,𝑠) is measurable;(2)for almost all 𝑥∈Ω, 𝑡↦𝐹(𝑥,𝑡,𝑠) and 𝑠↦𝐹(𝑥,𝑡,𝑠) are locally Lipschitz.

Lemma 3.1. Suppose H(F) and the following conditions hold:(f₁) there exists 𝛼∈𝐶₊(Ω) and 𝛼(𝑥)<𝑝^∗(𝑥), such that 𝑤₁≤𝑐₁1+|𝑡|^{𝛼(𝑥)−1}+|𝑠|^{𝛽(𝑥)(𝛼(𝑥)−1)/𝛼(𝑥)}, (3.3) for almost all 𝑥∈Ω, all 𝑡,𝑠∈ℝ and 𝑤₁∈𝜕_𝑡𝐹(𝑥,𝑡,𝑠);(f₂) there exists 𝛽∈𝐶₊(Ω) and 𝛽(𝑥)<𝑞^∗(𝑥), such that 𝑤₂≤𝑐₂1+|𝑡|^{(𝛼(𝑥)(𝛽(𝑥)−1))/(𝛽(𝑥))}+|𝑠|^{𝛽(𝑥)−1}, (3.4) for almost all 𝑥∈Ω, all 𝑡,𝑠∈ℝ and 𝑤₂∈𝜕_𝑠𝐹(𝑥,𝑡,𝑠);
then 𝜑(⋅,𝑣)(𝜑(𝑢,⋅)) is locally Lipschitz on 𝑊₀^1,𝑝(𝑥)(Ω)(𝑊₀^1,𝑞(𝑥)(Ω)).

Proof. We need only to prove that 𝜑(⋅,𝑣) is locally Lipschitz on 𝑊₀^1,𝑝(𝑥)(Ω).
By Φ(⋅,𝑣)∈𝐶¹(𝑊₀^1,𝑝(𝑥)(Ω),ℝ),we have 𝜑𝑢₁,𝑣−𝜑𝑢₂,𝑣=𝐽𝑢₁−𝐽𝑢₂=𝐽^′(𝑢)⋅𝑢₁−𝑢₂, (3.5) where𝑢=𝑡𝑢₁+(1−𝑡)𝑢₂,𝑡∈(0,1).
Let𝐵_𝑟={𝑢∈𝑊₀^1,𝑝(𝑥)(Ω)∶‖𝑢−𝑢₀‖_{𝑊01,𝑝(𝑥)(Ω)}≤𝑟}.
Note that 𝐵_𝑟 is 𝑤-compact. Then we obtain that there exists a positive constant 𝑀, such that ‖𝐽^′(𝑢)‖_{𝑊′−1,𝑝(𝑥)(Ω)}≤𝑀 with 1/𝑝(𝑥)+1/𝑝^′(𝑥)=1, for sufficiently small 𝑟.
Therefore, for any 𝑢₁,𝑢₂∈𝐵_𝑟, we have Φ𝑢₁,𝑣−Φ𝑢₂,𝑣=𝐽^′(𝑢)⋅𝑢₁−𝑢₂≤𝐽^′(𝑢)_{𝑊′−1,𝑝(𝑥)(𝑥)(Ω)}𝑢₁−𝑢_{2𝑊01,𝑝(𝑥)(Ω)}≤𝑀𝑢₁−𝑢_{2𝑊01,𝑝(𝑥)(Ω)},∀𝑣∈𝑊₀^1,𝑞(𝑥)(Ω). (3.6)
Fixing 𝑣∈𝑊₀^1,𝑞(𝑥)(Ω), by (f₁) and Lebourg mean value theorem, we have 𝐹𝑥,𝑢₁,𝑣−𝐹𝑥,𝑢₂,𝑣≤𝑐₁1+|𝑢|^{𝛼(𝑥)−1}+|𝑣|^{𝛽(𝑥)(𝛼(𝑥)−1)/𝛼(𝑥)}𝑢₁−𝑢₂. (3.7) Hence, _Ω𝐹𝑥,𝑢₁,𝑣𝑑𝑥−_Ω𝐹𝑥,𝑢₂,𝑣𝑑𝑥≤𝑐_1Ω𝑢₁−𝑢₂𝑑𝑥+𝑐_1Ω|𝑢|^{𝛼(𝑥)−1}𝑢₁−𝑢₂𝑑𝑥+𝑐_1Ω|𝑣|^{𝛽(𝑥)(𝛼(𝑥)−1)/𝛼(𝑥)}𝑢₁−𝑢₂≤𝑑𝑥𝑐₁𝑢₁−𝑢_2𝛼(𝑥)+𝑐₁|𝑢|^{𝛼(𝑥)−1}_{𝛼′(𝑥)}𝑢₁−𝑢_2𝛼(𝑥)+𝑐₁|𝑣|^{𝛽(𝑥)(𝛼(𝑥)−1)/𝛼(𝑥)}_{𝛼′(𝑥)}𝑢₁−𝑢_2𝛼(𝑥), (3.8) where1/𝛼^′(𝑥)+1/𝛼(𝑥)=1.
It is immediate that _Ω|𝑢|^{𝛼(𝑥)−1𝛼′(𝑥)}=_Ω|𝑢|^𝛼(𝑥)|𝑑𝑥≤𝑢|^𝛼+_𝛼(𝑥)≤𝑐‖𝑢‖^𝛼+,|𝑢|_𝛼(𝑥)|>1,𝑢|^𝛼+_𝛼(𝑥)≤𝑐‖𝑢‖^𝛼−,|𝑢|_𝛼(𝑥)<1,_Ω|𝑣|^{𝛽(𝑥)(𝛼(𝑥)−1)/𝛼(𝑥)𝛼′(𝑥)}=_Ω|𝑣|^𝛽(𝑥)|𝑣|𝑑𝑥≤^{𝛽+𝛽(𝑥)}≤𝑐‖𝑣‖^𝛽+,|𝑣|_𝛽(𝑥)|𝑣|>1,_{𝛽+𝛽(𝑥)}≤𝑐‖𝑣‖^𝛽−,|𝑣|_𝛽(𝑥)<1 (3.9) are bounded.
So, _Ω𝐹𝑥,𝑢₁,𝑣𝑑𝑥−_Ω𝐹𝑥,𝑢₂,𝑣𝑑𝑥≤𝑐‖𝑢₁−𝑢₂‖, (3.10) since 𝑊₀^1,𝑝(𝑥)(Ω)↪𝐿^𝛼(𝑥)(Ω) is a compact embedding.
Therefore, 𝜑(⋅,𝑣) is locally Lipschitz. Similarly, we can prove that 𝜑(𝑢,⋅) is locally Lipschitz.

Theorem 3.2. Suppose that H(F), (f₁) with 𝛼⁺<𝑝⁻, (f₂) with 𝛽⁺<𝑞⁻ and the following conditions (f₃)-(f₄) hold:(f₃) there exists 𝛾_𝑖∈𝐶(Ω)(𝑖=1,2) with 𝑝⁺<𝛾₁(𝑥)<𝑝^∗(𝑥),𝑞⁺<𝛾₂(𝑥)<𝑞^∗(𝑥) and 𝜇₁,𝜇₂∈𝐿^∞(Ω), such that limsup_{𝑡→0,𝑠→0}𝑤₁,𝑡|𝑡|^𝛾1(𝑥)<𝜇₁(𝑥),limsup_{𝑡→0,𝑠→0}𝑤₂,𝑠|𝑠|^𝛾2(𝑥)<𝜇₂(𝑥) (3.11) uniformly for almost all 𝑥∈Ω and all 𝑤₁∈𝜕_𝑡𝐹(𝑥,𝑡,𝑠),𝑤₂∈𝜕_𝑠𝐹(𝑥,𝑡,𝑠);(f₄) there exist 𝜉₀∈𝑅,𝜂₀∈𝑅,𝑥₀∈Ω and 𝑟₀>0, such that 𝐹𝑥,𝜉₀,𝜂₀>𝛿₀>0,a.e.𝑥∈𝐵_𝑟0𝑥₀, (3.12) where 𝐵_𝑟0(𝑥₀)∶={𝑥∈Ω∶|𝑥−𝑥₀|≤𝑟₀}⊂Ω.

Then there exists 𝜆_∗>0 such that, for each 𝜆>𝜆_∗, the problem (2) has at least two nontrivial solutions.

Proof. The proof is divided into four steps as follows.

Step 1. We will show that 𝜑 is coercive in the step.
Firstly, for almost all 𝑥∈Ω, by H(F)(2), 𝑡↦𝐹(𝑥,𝑡,𝑠) is differentiable almost everywhere on ℝ and we have 𝑑𝐹()𝑑𝑡𝑥,𝑡,𝑠∈𝜕_𝑡𝐹(𝑥,𝑡,𝑠). (3.13) Moreover, from (f₁), (f₂) and Young inequality, we can get that 𝐹(𝑥,𝑡,𝑠)=𝐹(𝑥,0,𝑠)+^𝑡₀𝑑𝐹()+𝑑𝑦𝑥,𝑦,𝑠𝑑𝑦=𝐹𝑥,0,0^𝑠₀𝑑𝐹()𝑑𝑧𝑥,0,𝑧𝑑𝑧+^𝑡₀𝑑𝐹𝑑𝑦𝑥,𝑦,𝑠𝑑𝑦≤𝑐₁𝑡|+|𝑡||^𝛼(𝑥)𝛼(𝑥)+|𝑠|^{𝛽(𝑥)(𝛼(𝑥)−1)/𝛼(𝑥)}|𝑡|+𝑐₂|𝑠|+|𝑡|^{𝛼(𝑥)(𝛽(𝑥)−1)/𝛽(𝑥)}|𝑠|+1𝛽(𝑥)|𝑠|^𝛽(𝑥)≤𝑐₁|𝑡|+|𝑡|^𝛼(𝑥)+(𝛼(𝑥)−1)|𝑠|^𝛽(𝑥)𝛼(𝑥)+|𝑡|^𝛼(𝑥)𝛽(𝑥)+𝑐₂|𝑠|+|𝑠|^𝛽(𝑥)+𝛽(𝑥)−1|𝑡|^𝛼(𝑥)𝛽(𝑥)+|𝑠|^𝛽(𝑥)𝛽(𝑥)≤𝑐₁|𝑡|+2|𝑡|^𝛼(𝑥)+|𝑠|^𝛽(𝑥)+𝑐₂|𝑠|+2|𝑠|^𝛽(𝑥)+|𝑡|^𝛼(𝑥), (3.14) for almost all 𝑥∈Ω and 𝑡,𝑠∈ℝ.
Note that 1<𝛼(𝑥)≤𝛼⁺<𝑝⁻<𝑝^∗(𝑥) and 1<𝛽(𝑥)≤𝛽⁺<𝑞⁻<𝑞^∗(𝑥), then, by Lemma 2.1, we have 𝑊₀^1,𝑝(𝑥)(Ω)↪𝐿^𝛼(𝑥)(Ω) and 𝑊₀^1,𝑞(𝑥)(Ω)↪𝐿^𝛽(𝑥)(Ω) (compact embedding). Furthermore, there exists 𝑐₃,𝑐₄>0 such that |𝑢|_𝛼(𝑥)≤𝑐₃‖𝑢‖ and |𝑣|_𝛽(𝑥)≤𝑐₄‖𝑣‖.
So, for any |𝑢|_𝛼(𝑥)>1 and ‖𝑢‖>1, _Ω|𝑢|^𝛼(𝑥)𝑑𝑥≤|𝑢|^𝛼+_𝛼(𝑥)≤𝑐^𝛼+₃‖𝑢‖^𝛼+ (3.15) and, for any |𝑣|_𝛽(𝑥)>1 and ‖𝑣‖>1, _Ω|𝑣|^𝛽(𝑥)𝑑𝑥≤|𝑣|^𝛽+_𝛽(𝑥)≤𝑐^𝛽+₄‖𝑣‖^𝛽+. (3.16)
By (3.14), (3.15), (3.31), the Hölder inequality and the Sobolev embedding theorem, we have 𝜑(𝑢,𝑣)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥+_Ω1𝑞(𝑥)|∇𝑣|^𝑞(𝑥)𝑑𝑥−𝜆_Ω𝐹()≥1𝑥,𝑢,𝑣𝑑𝑥𝑝⁺‖𝑢‖^𝑝−+1𝑞⁺‖𝑣‖^𝑞−−𝜆𝑐_1Ω|𝑢|𝑑𝑥−2𝑐₁𝑐^𝛼+₃‖𝑢‖^𝛼+−𝑐₁𝑐^𝛽+₄‖𝑣‖^𝛽+−𝜆𝑐_2Ω|𝑣|𝑑𝑥−2𝑐₂𝑐^𝛽+₄‖𝑣‖^𝛽+−𝑐₂𝑐^𝛼+₃‖𝑢‖^𝛼+≥1𝑝⁺‖𝑢‖^𝑝−+1𝑞⁺‖𝑣‖^𝑞−−2𝜆𝑐₁|1|_{𝛼′(𝑥)}|𝑢|_𝛼(𝑥)−2𝜆𝑐₁|1|_{𝛽′(𝑥)}|𝑣|_𝛽(𝑥)−2𝑐₁𝑐^𝛼+₃‖𝑢‖^𝛼+−𝑐₁𝑐^𝛽+₄‖𝑣‖^𝛽+−2𝑐₂𝑐^𝛽+₄‖𝑣‖^𝛽+−𝑐₂𝑐^𝛼+₃‖𝑢‖^𝛼+≥1𝑝⁺‖𝑢‖^𝑝−+1𝑞⁺‖𝑣‖^𝑞−−2𝜆𝑐₁𝑐|1|_{𝛼′(𝑥)}‖𝑢‖−2𝜆𝑐₁𝑐|1|_{𝛽′(𝑥)}‖𝑣‖−2𝑐₁𝑐^𝛼+₃‖𝑢‖^𝛼+−𝑐₁𝑐^𝛽+₄‖𝑣‖^𝛽+−2𝑐₂𝑐^𝛽+₄‖𝑣‖^𝛽+−𝑐₂𝑐^𝛼+₃‖𝑢‖^𝛼+⟶∞,as‖𝑢,𝑣‖⟶∞. (3.17)

Step 2. We will show that the 𝜑 is weakly lower semicontinuous.
Leting 𝑢_𝑛⇀𝑢 weakly in 𝑊₀^1,𝑝(𝑥)(Ω), 𝑣_𝑛⇀𝑣 weakly in 𝑊₀^1,𝑞(𝑥)(Ω) by Lemma 2.1(3), we obtain the following results:𝑊₀^1,𝑝(𝑥)(Ω)↪𝐿^𝑝(𝑥)(Ω); 𝑊₀^1,𝑞(𝑥)(Ω)↪𝐿^𝑞(𝑥)(Ω); 𝑢_𝑛→𝑢 in 𝐿^𝑝(𝑥)(Ω);𝑣_𝑛→𝑣 in 𝐿^𝑞(𝑥)(Ω);𝑢_𝑛→𝑢 for a.a. 𝑥∈Ω;𝑣_𝑛→𝑣 for a.a. 𝑥∈Ω;𝐹(𝑥,𝑢_𝑛(𝑥),𝑣_𝑛(𝑥))→𝐹(𝑥,𝑢(𝑥),𝑣(𝑥)) for a.a. 𝑥∈Ω.
By Fatou’s Lemma, limsup_𝑛→∞Ω𝐹𝑥,𝑢_𝑛(𝑥),𝑣_𝑛(𝑥)𝑑𝑥≤_Ω𝐹(𝑥,𝑢(𝑥),𝑣(𝑥))𝑑𝑥. (3.18)
Thus, liminf_𝑛→∞𝜑𝑢_𝑛,𝑣_𝑣=liminf_𝑛→∞Ω1𝑝(𝑥)∇𝑢_𝑛^𝑝(𝑥)𝑑𝑥+_Ω1𝑞(𝑥)∇𝑣_𝑛^𝑞(𝑥)𝑑𝑥−limsup_𝑛→∞Ω𝐹𝑥,𝑢_𝑛,𝑣_𝑛≥𝑑𝑥_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥+_Ω1𝑞(𝑥)|∇𝑣|^𝑞(𝑥)𝑑𝑥−𝜆_Ω𝐹()(𝑢)𝑥,𝑢,𝑣𝑑𝑥.=𝜑 (3.19)
Hence, by Theorem 2.6, we deduce that there exists a global minimizer (𝑢₀,𝑣₀)∈𝑊₀^1,𝑝(𝑥)(Ω)×𝑊₀^1,𝑞(𝑥)(Ω) such that 𝜑𝑢₀,𝑣₀=min_{()𝑢,𝑣∈𝑊01,𝑝(𝑥)(Ω)×𝑊01,𝑞(𝑥)(Ω)}𝜑(𝑢,𝑣). (3.20)

Step 3. We will show that there exists 𝜆_∗>0 such that for each 𝜆>𝜆_∗, 𝜑(𝑢₀,𝑣₀)<0.
By the condition (f₄), there exists 𝜉₀,𝜂₀∈ℝ such that 𝐹(𝑥,𝜉₀,𝜂₀)>𝛿₀>0, a.e. 𝑥∈𝐵_𝑟0(𝑥₀). It is clear that 0<𝑀₁∶=max_{|𝑡|≤𝜉0,|𝑠|≤𝜂0}𝑐₁|𝑡|+2|𝑡|^𝛼++|𝑠|^𝛽++𝑐₂|𝑠|+2|𝑠|^𝛽++|𝑡|^𝛼+,𝑐₁|𝑡|+2|𝑡|^𝛼−+|𝑠|^𝛽−+𝑐₂|𝑠|+2|𝑠|^𝛽−+|𝑡|^𝛼−<+∞. (3.21)
Now we denote 𝑡₀=𝑀₁𝛿₀+𝑀₁^1/𝑁,𝐾(𝑡)𝜉∶=max⁰𝑟₀(1−𝑡)^𝑝−,𝜉₀𝑟₀(1−𝑡)^𝑝+,𝜉₀𝑟₀(1−𝑡)^𝑞−,𝜉₀𝑟₀(1−𝑡)^𝑞+,𝜆_∗=max_{𝑡𝑡∈1,𝑡2}𝐾(𝑡)1−𝑡^𝑁𝛿₀𝑡^𝑁−𝑀₁1−𝑡^𝑁, (3.22) where 𝑡₀<𝑡₁<𝑡₂<1 and 𝛿₀ is given in the condition (f₄). A simple calculation shows that the function 𝑡↦𝛿₀𝑡^𝑁−𝑀₁(1−𝑡^𝑁) is positive whenever 𝑡>𝑡₀ and 𝛿₀𝑡^𝑁₀−𝑀₁(1−𝑡^𝑁₀)=0. Thus 𝜆_∗ is well defined and 𝜆_∗>0.
We will show that, for each 𝜆>𝜆_∗, the problem (𝑃) has two nontrivial solutions. In order to do this, for 𝑡∈[𝑡₁,𝑡₂], let us define 𝜂_𝑡(𝑥)=0,if𝑥∈Ω⧵𝐵_𝑟0𝑥₀,𝜉₀,if𝑥∈𝐵_𝑡𝑟0𝑥₀,𝜉₀𝑟₀()𝑟1−𝑡⁰−𝑥−𝑥₀,if𝑥∈𝐵_𝑟0𝑥₀⧵𝐵_𝑡𝑟0𝑥₀. (3.23)
By conditions (f₁) and (f₃) we have _Ω𝐹𝑥,𝜂_𝑡(𝑥),𝜂_𝑡(𝑥)𝑑𝑥=_{𝐵0𝑡𝑟(𝑥0)}𝐹𝑥,𝜂_𝑡(𝑥),𝜂_𝑡(𝑥)+𝑑𝑥_{𝐵𝑟0(𝑥0)⧵𝐵0𝑡𝑟(𝑥0)}𝐹𝑥,𝜂_𝑡(𝑥),𝜂_𝑡(𝑥)𝑑𝑥≥𝑤_𝑁𝑟^𝑁₀𝑡^𝑁𝛿₀−𝑀₁1−𝑡^𝑁𝑤_𝑁𝑟^𝑁₀=𝑤_𝑁𝑟^𝑁₀𝛿₀𝑡^𝑁−𝑀₁1−𝑡^𝑁. (3.24) Hence, for 𝑡∈[𝑡₁,𝑡₂], 𝜑𝜂_𝑡,𝜂_𝑡=_Ω1𝑝(𝑥)∇𝜂_𝑡^𝑝(𝑥)𝑑𝑥+_Ω1𝑞(𝑥)∇𝜂_𝑡^𝑞(𝑥)𝑑𝑥−𝜆_Ω𝐹𝑥,𝜂_𝑡(𝑥),𝜂_𝑡(𝑥≤1)𝑑𝑥𝑝⁻_Ω∇𝜂_𝑡^𝑝(𝑥)1𝑑𝑥+𝑞⁻_Ω∇𝜂_𝑡^𝑞(𝑥)𝑑𝑥−𝜆𝑤_𝑁𝑟^𝑁₀𝛿₀𝑡^𝑁−𝑀₁1−𝑡^𝑁𝜉≤max⁰𝑟₀(1−𝑡)^𝑝−,𝜉₀𝑟₀(1−𝑡)^𝑝+,𝜉₀𝑟₀(1−𝑡)^𝑞−,𝜉₀𝑟₀(1−𝑡)^𝑞+×𝑤_𝑁𝑟^𝑁₀1−𝑡^𝑁−𝜆𝑤_𝑁𝑟^𝑁₀𝛿₀𝑡^𝑁−𝑀₁1−𝑡^𝑁=𝑤_𝑁𝑟^𝑁₀𝐾(𝑡)1−𝑡^𝑁−𝜆𝛿₀𝑡^𝑁−𝑀₁1−𝑡^𝑁, (3.25) so that 𝜑(𝜂_𝑡,𝜂_𝑡)<0 whenever 𝜆>𝜆_∗.

Step 4. We will check the C-condition in the following.
Suppose {(𝑢_𝑛,𝑣_𝑛)}_𝑛≥1⊆𝑊₀^1,𝑝(𝑥)(Ω)×𝑊₀^1,𝑞(𝑥)(Ω) such that 𝜑(𝑢_𝑛,𝑣_𝑛)→𝑐 and (1+‖(𝑢_𝑛,𝑣_𝑛)‖)𝑚(𝑢_𝑛,𝑣_𝑛)→0. Let (𝑢^∗_𝑛,𝑣^∗_𝑛)∈𝜕𝜑(𝑢_𝑛,𝑣_𝑛) be such that 𝑚(𝑢_𝑛,𝑣_𝑛)=‖(𝑢^∗_𝑛,𝑣^∗_𝑛)‖_{(𝑊01,𝑝(𝑥)(Ω)×𝑊01,𝑞(𝑥)(Ω))∗}. The interpretation of (𝑢^∗_𝑛,𝑣^∗_𝑛)∈𝜕𝜑(𝑢_𝑛,𝑣_𝑛)is that 𝑢^∗_𝑛∈𝜕_𝑢𝜑(𝑢_𝑛,𝑣_𝑛) and 𝑣^∗_𝑛∈𝜕_𝑣𝜑(𝑢_𝑛,𝑣_𝑛). We know that 𝑢^∗_𝑛=−Δ_𝑝(𝑥)𝑢_𝑛−𝜆𝑤¹_𝑛,𝑣^∗_𝑛=−Δ_𝑞(𝑥)𝑣_𝑛−𝜆𝑤²_𝑛 (3.26) with 𝑤¹_𝑛∈𝜕_𝑢Ψ(𝑢_𝑛,𝑣_𝑛) and 𝑤²_𝑛∈𝜕_𝑣Ψ(𝑢_𝑛,𝑣_𝑛). From Chang [23] we know that 𝑤¹_𝑛∈𝐿^{𝛼′(𝑥)}(Ω) and 𝑤²_𝑛∈𝐿^{𝛽′(𝑥)}(Ω), where 𝛼^′(𝑥)=𝛼(𝑥)/(𝛼(𝑥)−1),𝛽′(𝑥)=𝛽(𝑥)/(𝛽(𝑥)−1).
Since 𝜑 is coercive, {𝑢_𝑛}_𝑛≥1, {𝑣_𝑛}_𝑛≥1 are bounded and passed to a subsequence, still denoting{𝑢_𝑛}_𝑛≥1 and {𝑣_𝑛}_𝑛≥1, we may assume that there exist 𝑢∈𝑊₀^1,𝑝(𝑥)(Ω), 𝑣∈𝑊₀^1,𝑞(𝑥)(Ω), such that 𝑢_𝑛⇀𝑢 weakly in 𝑊₀^1,𝑝(𝑥)(Ω) and 𝑣_𝑛⇀𝑣 weakly in 𝑊₀^1,𝑞(𝑥)(Ω). Next we will prove that 𝑢_𝑛→𝑢in𝑊₀^1,𝑝(𝑥)(Ω),𝑣_𝑛→𝑣in𝑊₀^1,𝑞(𝑥)(Ω). (3.27)
By 𝑊₀^1,𝑝(𝑥)(Ω)→𝐿^𝑝(𝑥)(Ω), 𝑊₀^1,𝑞(𝑥)(Ω)→𝐿^𝑞(𝑥)(Ω), we have 𝑢_𝑛→𝑢 in 𝐿^𝑝(𝑥)(Ω) and 𝑣_𝑛→𝑣 in 𝐿^𝑞(𝑥)(Ω). Moreover, since ‖(𝑢^∗_𝑛,𝑣^∗_𝑛)‖_{(𝑊01,𝑝(𝑥)(Ω)×𝑊01,𝑞(𝑥)(Ω))∗}→0, we get ‖𝑢^∗_𝑛‖_{(𝑊01,𝑝(𝑥)(Ω))∗}→0, ‖𝑣^∗_𝑛‖_{(𝑊01,𝑞(𝑥)(Ω))∗}→0, so |⟨𝑢^∗_𝑛,𝑢_𝑛⟩|≤𝜀_𝑛, |⟨𝑣^∗_𝑛,𝑣_𝑛⟩|≤𝜀_𝑛.
From (3.26), we have −Δ_𝑝(𝑥)𝑢_𝑛,𝑢_𝑛−𝑢−_Ω𝑤¹_𝑛𝑢_𝑛−𝑢𝑑𝑥≤𝜀_𝑛,∀𝑛≥1.−Δ_𝑞(𝑥)𝑣_𝑛,𝑣_𝑛−𝑣−_Ω𝑤²_𝑛𝑣_𝑛−𝑣𝑑𝑥≤𝜀_𝑛,∀𝑛≥1. (3.28) Moreover, ∫_Ω𝑤¹_𝑛(𝑢_𝑛−𝑢)𝑑𝑥→0 and ∫_Ω𝑤²_𝑛(𝑣_𝑛−𝑣)𝑑𝑥→0, since 𝑢_𝑛→𝑢 in 𝐿^𝑝(𝑥)(Ω), 𝑣_𝑛→𝑣 in 𝐿^𝑞(𝑥)(Ω), {𝑤¹_𝑛}_𝑛≥1 in 𝐿^{𝑝′(𝑥)}(Ω) and {𝑤²_𝑛}_𝑛≥1 in 𝐿^{𝑞′(𝑥)}(Ω) are bounded, where 1/𝑝(𝑥)+1/(𝑝^′(𝑥))=1,1/(𝑞(𝑥))+1/(𝑞^′(𝑥))=1. Therefore, limsup_𝑛→∞−Δ_𝑝(𝑥)𝑢_𝑛,𝑢_𝑛−𝑢≤0,limsup_𝑛→∞−Δ_𝑞(𝑥)𝑣_𝑛,𝑣_𝑛−𝑣≤0. (3.29)
From Lemma 2.5, we have 𝑢_𝑛→𝑢, 𝑣_𝑛→𝑣 as 𝑛→∞. Thus 𝜑 satisfies the nonsmooth C-condition.

Step 5. We will show that there exists another nontrivial weak solution of problem (𝑃).
From Lebourg Mean Value Theorem, we obtain 𝐹(𝑥,𝑡,𝑠)−𝐹(𝑥,0,𝑠)=𝑤₁,𝐹()()=𝑤,𝑡𝑥,0,𝑠−𝐹𝑥,0,0₂,𝑠 (3.30) for some 𝑤₁∈𝜕_𝑡𝐹(𝑥,𝜗𝑡,𝑠), 𝑤₂∈𝜕_𝑠𝐹(𝑥,0,𝜏𝑠), and 0<𝜗,𝜏<1. Thus by the condition (f₃), there exists 𝛽∈(0,1) such that |𝐹(𝑥,𝑡,𝑠)|≤𝑤₁,𝑡+𝑤₂,𝑠≤𝜇₁(𝑥)|𝑡|^𝛾1(𝑥)+𝜇₂(𝑥)|𝑠|^𝛾2(𝑥), (3.31) for all |𝑡|,|𝑠|<𝛽anda.e.𝑥∈Ω.
It follows from the conditions (f₁), (f₂) 1<𝛼⁻≤𝛼⁺<𝑝⁻≤𝑝⁺<𝛾₁(𝑥)<𝑝^∗(𝑥) and 1<𝛽⁻≤𝛽⁺<𝑞⁻≤𝑞⁺<𝛾₂(𝑥)<𝑝^∗(𝑥) that for all |𝑡|>𝛽, |𝑠|>𝛽 and a.e. 𝑥∈Ω, |𝐹(𝑥,𝑡,𝑠)|≤𝑐₁|𝑡|+2|𝑡|^𝛼(𝑥)+|𝑠|^𝛽(𝑥)+𝑐₂|𝑠|+2|𝑠|^𝛽(𝑥)+|𝑡|^𝛼(𝑥)≤𝑐₁𝛽^{𝛾1(𝑥)−1}+2𝑐₂𝛽^{𝛾1(𝑥)−𝛼(𝑥)}+𝑐₂𝛽^{𝛾1(𝑥)−𝛽(𝑥)}|𝑡|^𝛾1(𝑥)+𝑐₂𝛽^{𝛾2(𝑥)−1}+2𝑐₂𝛽^{𝛾2(𝑥)−𝛽(𝑥)}+𝑐₁𝛽^{𝛾1(𝑥)−𝛼(𝑥)}|𝑡|^𝛾2(𝑥)≤𝑐₁𝛽^𝛾+1−1+2𝑐₂𝛽^{𝛾+1−𝛼−}+𝑐₂𝛽^{𝛾+1−𝛽−}|𝑡|^𝛾1(𝑥)+𝑐₂𝛽^𝛾+2−1+2𝑐₂𝛽^{𝛾+2−𝛽−}+𝑐₁𝛽^{𝛾+1−𝛼−}|𝑡|^𝛾2(𝑥), (3.32) this together with (3.31) yields that, for all 𝑡∈ℝ and a.e. 𝑥∈Ω, |𝐹(𝑥,𝑡,𝑠)|≤𝜇₁(𝑥)+𝑐₁𝛽^𝛾+1−1+2𝑐₂𝛽^{𝛾+1−𝛼−}+𝑐₂𝛽^{𝛾+1−𝛽−}|𝑡|^𝛾1(𝑥)+𝜇₂(𝑥)+𝑐₂𝛽^𝛾+2−1+2𝑐₂𝛽^{𝛾+2−𝛽−}+𝑐₁𝛽^{𝛾+1−𝛼−}|𝑡|^𝛾2(𝑥)≤𝑐₃|t|^𝛾1(𝑥)+𝑐₄|𝑠|^𝛾2(𝑥), (3.33) for positive constants 𝑐₃,𝑐₄.
Note that 𝑝⁺<𝛾₁(𝑥)<𝑝^∗(𝑥), 𝑞⁺<𝛾₂(𝑥)<𝑞^∗(𝑥), then, by Lemma 2.1, we have 𝑊₀^1,𝑝(𝑥)(Ω)↪𝐿^𝛾1(𝑥)(Ω) and 𝑊₀^1,𝑞(𝑥)(Ω)↪𝐿^𝛾2(𝑥)(Ω). Furthermore, there exist 𝑐₅,𝑐₆ such that |𝑢|_𝛾1(𝑥)≤𝑐₅‖𝑢‖,∀𝑢∈𝑊₀^1,𝑝(𝑥)(Ω),|𝑣|_𝛾2(𝑥)≤𝑐₆‖𝑢‖,∀𝑣∈𝑊₀^1,𝑞(𝑥)(Ω). (3.34)
For all 𝜆>𝜆_∗, ‖(𝑢,𝑣)‖<1, |𝑢|_𝛾1(𝑥)<1 and |𝑣|_𝛾2(𝑥)<1, from (3.33) we have 𝜑(𝑢,𝑣)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥+_Ω1𝑞(𝑥)|∇𝑣|^𝑞(𝑥)𝑑𝑥−𝜆_Ω𝐹((𝑥)(𝑥≥𝑥,𝑢,𝑣))𝑑𝑥_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥+_Ω1𝑞(𝑥)|∇𝑣|^𝑞(𝑥)𝑑𝑥−𝜆𝑐_3Ω|𝑢(𝑥)|^𝛾1(𝑥)𝑑𝑥−𝜆𝑐_4Ω|𝑣(𝑥)|^𝛾2(𝑥)≥1𝑑𝑥𝑝⁺‖𝑢‖^𝑝++1𝑞⁺‖𝑣‖^𝑞+−𝜆𝑐₃𝑐^𝛾−1₅‖𝑢‖^𝛾−1+𝑐₄𝑐^𝛾−2₆‖𝑢‖^𝛾−2. (3.35)
So, for 𝜌>0 small enough, there exists a 𝜈>0 such that 𝜑(𝑢,𝑣)>𝜈,for‖𝑢,𝑣‖=𝜌 (3.36) and ‖(𝑢₀,𝑣₀)‖>𝜌. So by the Nonsmooth Mountain Pass Theorem (cf. Theorem 2.7), we can get (𝑢₁,𝑣₁)∈𝑊₀^1,𝑝(𝑥)(Ω)×𝑊₀^1,𝑞(𝑥)(Ω)  which satisfies 𝜑𝑢₁,𝑣₁=𝑐>0,𝑚𝑢₁,𝑣₁=0. (3.37)
Therefore, (𝑢₁,𝑣₁) is another nontrivial solution of problem (𝑃).

Remark 3.3. Let 𝑝⁻>𝛼⁺, 𝑞⁻>𝛽⁺ and consider the following nonsmooth locally Lipschitz function: 𝐹()=𝑡𝑥,𝑡,𝑠^𝛾1(𝑥)+𝑠^𝛾2(𝑥),𝑡∈(0,𝛿)or(),|𝑠∈0,𝛿max|𝑡−𝛿^𝜃(𝑥),|𝑡−𝛿|^𝛼(𝑥)+|𝛿|^𝛾1(𝑥)+max|𝑠−𝛿|^𝜃(𝑥),|𝑠−𝛿|^𝛽(𝑥)+|𝛿|^𝛾2(𝑥),𝑡∈[𝛿,+∞)or[),(]𝑠∈𝛿,+∞0,𝑡∈−∞,0or𝑠∈(−∞,0], (3.38) where 0<𝛿<1, 𝜃⁻>1, 𝜃⁺<𝛼⁻ and 𝜃⁺<𝛽⁻.
Obvious, 𝑡↦𝐹(𝑥,𝑡,𝑠) and 𝑠↦𝐹(𝑥,𝑡,𝑠) are locally Lipschitz. Then, 𝜕_𝑡𝐹()=𝛾𝑥,𝑡,𝑠¹(𝑥)𝑡^{𝛾1(𝑥)−1}(),𝜃(𝑥),𝑡∈0,𝛿)(𝑡−𝛿^{𝜃(𝑥)−1}(),𝛼(𝑥),𝑡∈𝛿,1+𝛿)(𝑡−𝛿^{𝛼(𝑥)−1},𝑡∈(1+𝛿,+∞),0,𝛾₁(𝑥)𝛿^{𝛾1(𝑥)−1}[𝜃(𝑥)(𝑥(],𝜕,𝑡=𝛿,,𝛼)],𝑡=1+𝛿,0,𝑡∈−∞,0_𝑠𝐹()=𝛾𝑥,𝑡,𝑠²(𝑥)𝑠^{𝛾2(𝑥)−1}(),𝜃(𝑥),𝑠∈0,𝛿)(𝑠−𝛿^{𝜃(𝑥)−1}(),𝛽(𝑥),𝑠∈𝛿,1+𝛿)(𝑡−𝛿^{𝛽(𝑥)−1},𝑠∈(1+𝛿,+∞),0,𝛾₁(𝑥)𝛿^{𝛾1(𝑥)−1}𝜃(𝑥)(𝑥(].,𝑠=𝛿,,𝛽),𝑠=1+𝛿,0,𝑠∈−∞,0 (3.39)
Hence, for any 𝑤₁∈𝜕_𝑡𝐹(𝑥,𝑡,𝑠) and 𝑤₂∈𝜕_𝑠𝐹(𝑥,𝑡,𝑠), we have 𝑤₁≤𝛾₁(𝑥)𝑡^{𝛼(𝑥)−1}𝑡^{𝛾1(𝑥)−𝛼(𝑥)}≤𝛾⁺₁|𝑡|^{𝛼(𝑥)−1}(),𝜃(𝑥),𝑡∈0,𝛿)(𝑡−𝛿^{𝜃(𝑥)−1}<𝜃⁺<𝜃⁺1𝛿^𝛼+−1|𝑡|^{𝛼(𝑥)−1}(),𝛼(𝑥),𝑡∈𝛿,1+𝛿)(𝑡−𝛿^{𝛼(𝑥)−1}≤𝛼⁺|𝑡|^{𝛼(𝑥)−1}(),𝛾,𝑡∈1+𝛿,+∞₁(𝑥)𝛿^{𝛾1(𝑥)−1}=𝛾₁(𝑥)𝛿^{𝛼(𝑥)−1}𝛿^{𝛾1(𝑥)−𝛼(𝑥)}≤𝛾⁺𝛿^{𝛼(𝑥)−1}[𝜃(𝑥)(𝑥,𝑡=𝛿,,𝛼)]≤𝛼⁺(1+𝛿)^{𝛼(𝑥)−1}(],𝑤,𝑡=1+𝛿,0,𝑡∈−∞,0₂≤𝛾₂(𝑥)𝑠^{𝛽(𝑥)−1}𝑠^{𝛾2(𝑥)−𝛽(𝑥)}≤𝛾⁺₂|𝑠|^{𝛽(𝑥)−1}(),𝜃(𝑥),𝑠∈0,𝛿)(𝑠−𝛿^{𝜃(𝑥)−1}<𝜃⁺<𝜃⁺1𝛿^𝛽+−1|𝑠|^{𝛽(𝑥)−1}(),𝛽(𝑥),𝑠∈𝛿,1+𝛿)(𝑠−𝛿^{𝛽(𝑥)−1}≤𝛽⁺|𝑠|^{𝛽(𝑥)−1}(),𝛾,𝑠∈1+𝛿,+∞₂(𝑥)𝛿^{𝛾2(𝑥)−1}=𝛾₂(𝑥)𝛿^{𝛽(𝑥)−1}𝛿^{𝛾2(𝑥)−𝛽(𝑥)}≤𝛾⁺𝛿^{𝛽(𝑥)−1}𝜃(𝑥)(𝑥,𝑠=𝛿,,𝛽)≤𝛽⁺(1+𝛿)^{𝛽(𝑥)−1}(].,𝑠=1+𝛿,0,𝑠∈−∞,0 (3.40)
Therefore, 𝑤₁≤𝛾⁺₁+𝛼⁺+𝜃⁺1𝛿^(𝛼+−1)|𝑡|^{𝛼(𝑥)−1},∀𝑤₁∈𝜕_𝑡𝐹(),𝑤𝑥,𝑡,𝑠₂≤𝛾⁺₂+𝛽⁺+𝜃⁺1𝛿^(𝛽+−1)|𝑡|^{𝛽(𝑥)−1},∀𝑤₂∈𝜕_𝑠𝐹(𝑥,𝑡,𝑠),limsup_{𝑡→0,𝑠→0}<𝑤₁,𝑡>|𝑡|^𝛾1(𝑥)=lim_𝑡→0𝛾₁(𝑥)𝑡^𝛾1(𝑥)𝑡^𝛾1(𝑥),𝑡>00,𝑡≤0≤𝛾₁(𝑥),limsup_{𝑡→0,𝑠→0}<𝑤₂,𝑠>|𝑠|^𝛾2(𝑥)=lim_𝑠→0𝛾₂(𝑥)𝑠^𝛾2(𝑥)𝑠^𝛾2(𝑥),𝑠>00,𝑠≤0≤𝛾₂(𝑥), (3.41) uniformly for almost all 𝑥∈Ω, all 𝑤₁∈𝜕_𝑡𝐹(𝑥,𝑡,𝑠) and 𝑤₂∈𝜕_𝑠𝐹(𝑥,𝑡,𝑠).
Thus far the results involved potential functions exhibiting 𝑝(𝑥)-sublinear. The next theorem concerns problems where the potential function is 𝑝(𝑥)-superlinear.

Theorem 3.4. Suppose that H(F), (f₁) with 𝛼⁻>𝑝⁺, (f₂) with 𝛽⁻>𝑞⁺, (f₃), (f₄) and the following condition (f₅) hold. (f₅)For almost all 𝑥∈Ω and all 𝑡,𝑠∈ℝ, one has 𝐹(𝑥,𝑡,𝑠)≤𝜅(𝑥)with𝜅∈𝐿^𝛾3(𝑥)(Ω),1≤𝛾₃(𝑥)<min{𝑝⁻,𝑞⁻}.
Then there exists a 𝜆_∗>0 such that, for each 𝜆>𝜆_∗, the problem (𝑃) has at least two nontrivial solutions.

Proof. The steps are similar to those of Theorem 3.2. In fact,we only need to modify Step 1 and Step 5 as follows: Step 6 shows, that 𝜑 is coercive under the condition (f₅); Step 7 shows, that there exists second nontrivial solution under the conditions (f₁), (f₂), and (f₃). Then from Steps 6, 2, 3, 4, and 7 above, the problem (𝑃) has at least two nontrivial solutions.

Step 6. By (f₅), for all (𝑢,𝑣)∈𝑊₀^1,𝑝(𝑥)(Ω)×𝑊₀^1,𝑞(𝑥)(Ω), ‖(𝑢,𝑣)‖>1, we have 𝜑(𝑢,𝑣)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥+_Ω1𝑞(𝑥)|∇𝑣|^𝑞(𝑥)𝑑𝑥−𝜆_Ω𝐹((𝑥)(𝑥≥1𝑥,𝑢,𝑣))𝑑𝑥𝑝⁺‖𝑢‖^𝑝++1𝑞⁺‖𝑣‖^𝑞+−𝜆_Ω𝜅(𝑥)𝑑𝑥⟶∞⟶∞,as‖(𝑢,𝑣)‖→∞. (3.42)

Step 7. Because of hypothesis (f₁), (f₂) and mean value theorem for locally Lipschitz functions, we have 𝐹(𝑥,𝑡,𝑠)≤𝑐₁|𝑡|+2|𝑡|^𝛼(𝑥)+|𝑠|^𝛽(𝑥)+𝑐₂|𝑠|+2|𝑠|^𝛽(𝑥)+|𝑡|^𝛼(𝑥)≤𝑐₁𝑡𝛽^{𝛼(𝑥)−1}|𝑡|+2|𝑡|^𝛼(𝑥)+|𝑠|^𝛽(𝑥)+𝑐₂𝑠𝛽^{𝛽(𝑥)−1}|𝑠|+2|𝑠|^𝛽(𝑥)+|𝑡|^𝛼(𝑥)≤𝑐₁1𝛽^𝛼+−1|𝑡|^𝛼(𝑥)+2|𝑡|^𝛼(𝑥)+|𝑠|^𝛽(𝑥)+𝑐₂1𝛽^𝛽+−1|𝑡|^𝛽(𝑥)+2|𝑠|^𝛽(𝑥)+|𝑡|^𝛼(𝑥)=𝑐₇|𝑡|^𝛼(𝑥)+𝑐₈|𝑡|^𝛽(𝑥) (3.43) for a.e. 𝑥∈Ω, all |𝑡|≥𝛽, |𝑠|≥𝛽 with 𝑐₇,𝑐₈>0.
Combining (3.31) and (3.43), it follows that |𝐹(𝑥,𝑡,𝑠)|≤𝜇₁(𝑥)|𝑡|^𝛾1(𝑥)+𝑐₇|𝑡|^𝛼(𝑥)+𝜇₂(𝑥)|𝑠|^𝛾2(𝑥)+𝑐₈|𝑠|^𝛼(𝑥) (3.44) for a.e. 𝑥∈Ω and all 𝑡,𝑠∈ℝ.
Thus, for all 𝜆>𝜆_∗, ‖(𝑢,𝑣)‖<1, |𝑢|_𝛾1(𝑥)<1, |𝑢|_𝛼(𝑥)<1|𝑣|_𝛾2(𝑥)<1 and |𝑣|_𝛽(𝑥)<1, we have 𝜑(𝑢,𝑣)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥+_Ω1𝑞(𝑥)|∇𝑣|^𝑞(𝑥)𝑑𝑥−𝜆_Ω𝐹((𝑥)(𝑥≥1𝑥,𝑢.𝑣))𝑑𝑥𝑝⁺‖𝑢‖^𝑝++1𝑞⁺‖𝑣‖^𝑞+−𝜆_Ω𝜇₁(𝑥)|𝑢|^𝛾1(𝑥)𝑑𝑥−𝜆𝑐_7Ω|𝑢|^𝛼(𝑥)𝑑𝑥−𝜆_Ω𝜇₂(𝑥)|𝑣|^𝛾2(𝑥)𝑑𝑥−𝜆𝑐_8Ω|𝑣|^𝛽(𝑥)≥1𝑑𝑥𝑝⁺‖𝑢‖^𝑝++1𝑞⁺‖𝑣‖^𝑞+−𝜆𝑐₉‖𝑢‖^𝛾−1−𝜆𝑐₇‖𝑢‖^𝛼−−𝜆𝑐₁₀‖𝑣‖^𝛾−2−𝜆𝑐₈‖𝑣‖^𝛽−. (3.45)
So, for 𝜌>0 small enough, there exists a 𝜈>0 such that 𝜑(𝑢,𝑣)>𝜈,for‖𝑢,𝑣‖=𝜌 (3.46) and ‖(𝑢₀,𝑣₀)‖>𝜌. Arguing as in proof of Step 4 of Theorem 3.2, we conclude that 𝜑 satisfies the nonsmooth C-condition. So by the Nonsmooth Mountain Pass Theorem (cf. Theorem 2.7), we can get (𝑢₁,𝑣₁)∈𝑊₀^1,𝑝(𝑥)(Ω)×𝑊₀^1,𝑞(𝑥)(Ω)  which satisfies 𝜑𝑢₁,𝑣₁=𝑐>0,𝑚𝑢₁,𝑣₁=0. (3.47)
Therefore, (𝑢₁,𝑣₁) is second nontrivial of problem (𝑃).

Remark 3.5. Consider the following nonsmooth locally Lipschitz function: 𝐹()=−|𝑡|𝑥,𝑡,𝑠^𝛾1(𝑥)−|𝑠|^𝛾2(𝑥),|𝑡|≤1or|𝑠|(𝜋|𝑡(𝜋|𝑠,|𝑡|≤1,cos|)+cos|)>1;or|𝑠|>1, (3.48) In the following, we will show that 𝐹(𝑥,𝑡,𝑠) satisfies hypotheses H(F) and (f₁)–(f₅).
It is clear that 𝐹(𝑥,0,0)=0 for a.e. 𝑥∈Ω,thenhypotheses H(F) is satisfied. A direct verification shows that conditions (f₄) and (f₅) are satisfied. Note that 𝜕_𝑡𝐹(𝑥,𝑡,𝑠)=−𝛾₁(𝑥)𝑡^{𝛾1(𝑥)−1}𝛾,0≤𝑡<1,₁(𝑥)(−𝑡)^{𝛾1(𝑥)−1},−1<𝑡≤0,−𝛾₁(𝑥),0,𝑡=1,0,𝛾₁(𝑥{(,|𝑡|𝜕),𝑡=−1,−𝜋sin𝜋𝑡)}>1,_𝑠𝐹(𝑥,𝑡,𝑠)=−𝛾₂(𝑥)𝑠^{𝛾2(𝑥)−1}𝛾,0≤𝑠<1,₂(𝑥)(−𝑠)^{𝛾1(𝑥)−1},−1<𝑠≤0,−𝛾₂(𝑥),0,𝑠=1,0,𝛾₂(𝑥{(,|𝑠|),𝑠=−1,−𝜋sin𝜋𝑠)}>1. (3.49)
So, 𝑤₁≤𝛾₁(𝑥)+𝜋|𝑡|^{𝛾1(𝑥)−1},∀𝑤₁∈𝜕_𝑡𝐹(),𝑤𝑥,𝑡,𝑠₁≤𝛾₂(𝑥)+𝜋|𝑠|^{𝛾2(𝑥)−1},∀𝑤₂∈𝜕_𝑠𝐹(𝑥,𝑡,𝑠),limsup_{𝑡→0,𝑠→0}<𝑤₁,𝑡>|𝑡|^𝛾1(𝑥)=lim_𝑡→0−𝛾₁(𝑥)𝑡^𝛾1(𝑥)𝑡^𝛾1(𝑥),𝑡>0lim_𝑡→0−𝛾₁(𝑥)(−𝑡)^𝛾1(𝑥)(−𝑡)^𝛾1(𝑥),𝑡≤0=−𝛾₁(𝑥)≤𝛾₁(𝑥),limsup_{𝑡→0,𝑠→0}<𝑤₂,𝑠>|𝑠|^𝛾2(𝑥)=lim_𝑡→0−𝛾₂(𝑥)𝑠^𝛾2(𝑥)𝑠^𝛾2(𝑥),𝑠>0lim_𝑡→0−𝛾₂(𝑥)(−𝑠)^𝛾2(𝑥)(−𝑠)^𝛾2(𝑥),𝑠≤0=−𝛾₂(𝑥)≤𝛾₂(𝑥) (3.50) which shows that assumptions (f₁), (f₂), and (f₃) are fulfilled.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (nos. 11126286, 11001063, 10971043), the Fundamental Research Funds for the Central Universities (no. 2012), China Postdoctoral Science Foundation Funded Project (no. 20110491032), and the National Natural Science Foundation of Heilongjiang Province (no. A200803).