Three Solutions for Inequalities Dirichlet Problem Driven by 𝑝(𝑥)-Laplacian-Like

Abstract

A class of nonlinear elliptic problems driven by 𝑝(𝑥)-Laplacian-like with a nonsmooth locally Lipschitz potential was considered. Applying the version of a nonsmooth three-critical-point theorem, existence of three solutions of the problem is proved.

1. Introduction

Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, the existence of multiple solutions of the problems with discontinuous nonlinearities has been widely investigated in recent years. In 1981, Chang [1] extended the variational methods to a class of nondifferentiable functionals and directly applied the variational methods for nondifferentiable functionals to prove some existence theorems for PDE with discontinuous nonlinearities. Soon thereafter, Kourogenis and Papageorgiou [2] extend the nonsmooth critical point theory of Chang [1], by replacing the compactness and the boundary conditions. In [3], by using the Ekeland variational principle and a deformation theorem, Kandilakis et al. obtained the local linking theorem for locally Lipschitz functions. In the celebrated work [4], Ricceri elaborated a Ricceri-type variational principle for Gateaux differentiable functionals. Later, Marano and Motreanu [5] extended Ricceri's result to a large class of nondifferentiable functionals and gave an application to a Neumann-type problem involving the 𝑝-Laplacian with discontinuous nonlinearities.

In this paper, we consider a nonlinear elliptic problem driven by 𝑝(𝑥)-Laplacian-like with a nonsmooth locally Lipschitz potential (hemivariational inequality): ||−div1+∇𝑢^𝑝(𝑥)1+|∇𝑢|^2𝑝(𝑥)|∇𝑢|^{𝑝(𝑥)−2}∇𝑢∈𝜆𝜕𝐹(𝑥,𝑢),a.e.inΩ,𝑢=0,on𝜕Ω, (P) where Ω⊂ℝ^𝑁 is a bounded domain with 𝐶¹-boundary 𝜕Ω. 𝑝∈𝐶(Ω), 2≤𝑁<𝑝⁻∶=inf_𝑥∈Ω𝑝(𝑥)≤𝑝⁺∶=sup_𝑥∈Ω𝑝(𝑥)<+∞,  𝐹∈𝐶(Ω×ℝ), and 𝐹∶Ω×ℝ→ℝ is a locally Lipschitz with respect to the second variable. By 𝜕𝐹(𝑥,𝑢), we denote the generalized subdifferential of the locally Lipschitz function 𝑢→𝐹(𝑥,𝑢). Our goal is to establish the same results under different assumptions.

The study of differential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, and so forth (see [6, 7]). The study on variable exponent problems attracts more and more interest in recent years. Many results have been obtained on this kind of problems, for example, [8–14]. Neumann-type problems involving the 𝑝(𝑥)-Laplacian have been studied, for instance, in [15–18].

Recently, Rodrigues [19] has considered the existence of nontrivial solution for the Dirichlet problem involving the 𝑝(𝑥)-Laplacian-like of the type ||−div1+∇𝑢^𝑝(𝑥)1+|∇𝑢|^2𝑝(𝑥)|∇𝑢|^{𝑝(𝑥)−2}∇𝑢=𝜆𝑓(𝑥,𝑢),a.e.inΩ,𝑢=0,on𝜕Ω, (1) where Ω⊂ℝ^𝑁 is a bounded domain with smooth boundary 𝜕Ω, 𝑝∈𝐶(Ω) with 𝑝(𝑥)>2, for all 𝑥∈Ω, and 𝑓∶Ω×ℝ→ℝ satisfies the Caratheodory condition. We emphasize that, in our approach, no continuity hypothesis will be required for the function 𝑓 with respect to the second argument. So, (P) need not have a solution. To avoid this situation, we consider such function 𝑓(𝑥,⋅) which is locally essentially bounded and fill the discontinuity gap of 𝑓(𝑥,⋅), replacing 𝑓 by the interval [𝑓₁,𝑓₂], where 𝑓₁(𝑥,𝑡)∶=lim_𝑠→0+essinf_{||𝑠−𝑡<𝛿}𝑓(),𝑓𝑥,𝑠₂(𝑥,𝑡)∶=lim_𝑠→0+esssup_{||𝑠−𝑡<𝛿}𝑓(𝑥,𝑠). (2) On the other hand, it is well known that if 𝐹(𝑥,𝑢)=∫^𝑢₀𝑓(𝑥,𝑡)𝑑𝑡, then 𝐹 become locally Lipschitz and 𝜕𝐹(𝑥,𝑢)=[𝑓₁(𝑥,𝑢),𝑓₂(𝑥,𝑢)] (see [1, 20]).

The aim of the present paper is to establish a three-solution theorem for the nonlinear elliptic problem driven by 𝑝(𝑥)-Laplacian-like with nonsmooth potential (see Theorem 6) by using a consequence (see Theorem 4) of the three-critical-point theorem established firstly by Marano and Motreanu in [20], which is a non-smooth version of Ricceri's three-critical-point theorem (see [21]). The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the generalized gradient of the locally Lipschitz function. In Section 3, we give the main result of this paper and use the non-smooth three-critical-point theorem to prove it.

2. Preliminary

In order to discuss problem (P), we need some theories on 𝑊₀^1,𝑝(𝑥)(Ω) and the generalized gradient of the locally Lipschitz function. Firstly we state some basic properties of space 𝑊₀^1,𝑝(𝑥)(Ω) which will be used later (for details, see [10–12]). Denote by 𝑆(Ω) the set of all measurable real functions defined on Ω. Two functions in 𝑆(Ω) are considered as the same element of 𝑆(Ω) when they are equal almost everywhere.

Put 𝐶₊(Ω)={𝑝∈𝐶(Ω)∶𝑝(𝑥)>1,∀𝑥∈Ω}.

If 𝑝∈𝐶(Ω), then write 𝐿^𝑝(𝑥)(Ω)=𝑢∈𝑆(Ω)∶_Ω|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥<+∞, (3) with the norm |𝑢|_𝐿^𝑝(𝑥)_(Ω)=|𝑢|_𝑝(𝑥)=inf{𝜆>0∶∫_Ω|𝑢(𝑥)/𝜆|^𝑝(𝑥)𝑑𝑥≤1}, and 𝑊^1,𝑝(𝑥)(Ω)=𝑢∈𝐿^𝑝(𝑥)(Ω)∶|∇𝑢|∈𝐿^𝑝(𝑥)(Ω), (4) with the norm ‖𝑢‖_𝑊^1,𝑝(𝑥)_(Ω)=|𝑢|_𝐿^𝑝(𝑥)_(Ω)+|∇𝑢|_𝐿^𝑝(𝑥)_(Ω). Denote by 𝑊₀^1,𝑝(𝑥)(Ω) the closure of 𝐶^∞₀(Ω) in 𝑊^1,𝑝(𝑥)(Ω).

We remember that the variable exponent Lebesgue spaces are separable and reflexive Banach spaces. Denote by 𝐿^𝑞(𝑥)(Ω) the conjugate Lebesgue space of 𝐿^𝑝(𝑥)(Ω) with 1/𝑝(𝑥)+1/𝑞(𝑥)=1; then the Hölder-type inequality _Ω||1𝑢𝑣𝑑𝑥≤𝑝⁻+1𝑞⁻|𝑢|_𝑝(𝑥)|𝑣|_𝑞(𝑥),𝑢∈𝐿^𝑝(𝑥)(Ω),𝑣∈𝐿^𝑞(𝑥)(Ω) (5) holds. Furthermore, if we define the mapping 𝜌∶𝐿^𝑝(𝑥)(Ω)→ℝ by 𝜌(𝑢)=_Ω|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥, (6) then the following relations hold: |𝑢|_𝑝(𝑥)>1⟹|𝑢|^𝑝−_𝑝(𝑥)≤𝜌(𝑢)≤|𝑢|^𝑝+_𝑝(𝑥),|𝑢|_𝑝(𝑥)<1⟹|𝑢|^𝑝+_𝑝(𝑥)≤𝜌(𝑢)≤|𝑢|^𝑝−_𝑝(𝑥). (7)

Proposition 1 (see [12]). In 𝑊₀^1,𝑝(𝑥)(Ω) Poincare's inequality holds; that is, there exists a positive constant 𝐶₀ such that |𝑢|_𝑝(𝑥)≤𝐶₀|∇𝑢|_𝑝(𝑥),∀𝑢∈𝑊₀^1,𝑝(𝑥)(Ω). (8) So |∇𝑢|_𝑝(𝑥) is an equivalent norm in 𝑊₀^1,𝑝(𝑥)(Ω).

We will use the equivalent norm in the following discussion and write ‖𝑢‖=|∇𝑢|_𝑝(𝑥) for simplicity.

Proposition 2 (see [10]). 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.
Consider the following function: 𝐽(𝑢)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+1+|∇𝑢|^2𝑝(𝑥)𝑑𝑥,𝑢∈𝑊₀^1,𝑝(𝑥)(Ω). (9) We know that (see [1]).
If one denotes 𝐴=𝐽^′∶𝑊₀^1,𝑝(𝑥)(Ω)→(𝑊₀^1,𝑝(𝑥)(Ω))^∗, then ⟨𝐴(𝑢)⟩=,𝑣_Ω|∇𝑢|^{𝑝(𝑥)−2}+|∇𝑢|^{2𝑝(𝑥)−2}1+|∇𝑢|^2𝑝(𝑥)(∇𝑢,∇𝑣)_ℝ^𝑁𝑑𝑥, (10) for all 𝑢,𝑣∈𝑊₀^1,𝑝(𝑥)(Ω).

Proposition 3 (see [19]). Set 𝑋=𝑊₀^1,𝑝(𝑥)(Ω); 𝐴 is as shown, then(1)  𝐴∶𝑋→𝑋^∗  is a convex, bounded previously; and strictly monotone operator; (2)   𝐴∶𝑋→𝑋^∗  is a mapping of type (𝑆)₊; that is, 𝑢−→_𝑛^𝑤𝑢  in 𝑋  and limsup_𝑛→∞⟨𝐴(𝑢_𝑛),𝑢_𝑛−𝑢⟩≤0  implies 𝑢_𝑛→𝑢  in 𝑋; (3)  𝐴∶𝑋→𝑋^∗  is a homeomorphism.
Let (𝑋,‖⋅‖) be a real Banach space, and let 𝑋^∗ be its topological dual. A function 𝑓∶𝑋→ℝ is called locally Lipschitz if each point 𝑢∈𝑋 possesses a neighborhood Ω_𝑢 such that |𝑓(𝑢₁)−𝑓(𝑢₂)|≤𝐿‖𝑢₁−𝑢₂‖ for all 𝑢₁,𝑢₂∈Ω_𝑢, for a positive constant 𝐿 depending on Ω_𝑢. The generalized directional derivative of 𝑓 at the point 𝑢∈𝑋 in the direction ℎ∈𝑋 is 𝑓⁰(𝑢;ℎ)=limsup_{𝑣→𝑢;𝑡↓0}𝑓(𝑣+𝑡ℎ)−𝑓(𝑣)𝑡. (11)
The generalized gradient of 𝑓 at 𝑢∈𝑋 is defined by 𝜕𝑓(𝑢)=𝑢^∗∈𝑋^∗∶𝑢^∗,ℎ≤𝑓⁰(𝑢;ℎ)∀ℎ∈𝑋, (12) which is a nonempty, convex, and 𝑤^∗-compact subset of 𝑋, where ⟨⋅,⋅⟩ is the duality pairing between 𝑋^∗ and 𝑋. One says that 𝑢∈𝑋 is a critical point of 𝑓 if 0∈𝜕𝑓(𝑢).

For further details, we refer the reader to the work of Chang [1].

Finally, for proving our results in the next section, we introduce the following theorem.

Theorem 4 (see [22, 23]). Let 𝑋 be a separable and reflexive real Banach space, and let Φ,Ψ∶𝑋→ℝ be two locally Lipschitz functions. Assume that there exists 𝑢₀∈𝑋 such that Φ(𝑢₀)=Ψ(𝑢₀)=0 and Φ(𝑢)≥0 for every 𝑢∈𝑋 and that there exist 𝑢₁∈𝑋 and 𝑟>0 such that(1)𝑟<Φ(𝑢₁); (2)sup_{Φ(𝑢)<𝑟}Ψ(𝑢)<𝑟(Ψ(𝑢₁)/Φ(𝑢₁)), and further, one assumes that function Φ−𝜆Ψ is sequentially lower semicontinuous and satisfies the (PS)-condition;(3)lim_{‖𝑢‖→∞}(Φ(𝑢)−𝜆Ψ(𝑢))=+∞ for every 𝜆∈[0,𝑎], where 𝑎=ℎ𝑟𝑟Ψ𝑢₁/Φ𝑢₁−sup_{Φ(𝑢)<𝑟}Ψ(𝑢),withℎ>1. (13)Then, there exist an open interval Λ₁⊆[0,𝑎] and a positive real number 𝜎 such that, for every 𝜆∈Λ₁, the function Φ(𝑢)−𝜆Ψ(𝑢) admits at least three critical points whose norms are less than 𝜎.

3. Existence Results

In this part, we will prove that there exist three solutions for problem (P) under certain conditions.

Definition 5. We say that 𝐼 satisfies (PS)_𝑐-condition if any sequence {𝑢_𝑛}⊂𝑊₀^1,𝑝(𝑥)(Ω), such that 𝐼(𝑢_𝑛)→𝑐 and 𝑚(𝑢_𝑛)→0, as 𝑛→+∞, has a strongly convergent subsequence, where 𝑚(𝑢_𝑛)=inf{‖𝑢^∗‖_𝑋^∗∶𝑢^∗∈𝜕𝐼(𝑢_𝑛)}.
By a solution of (P), we mean a function 𝑢∈𝑊₀^1,𝑝(𝑥)(Ω) to which there corresponds a mapping Ω∋𝑥→𝑤(𝑥) with 𝑤(𝑥)∈𝜕𝐹(𝑥,𝑢) for almost every 𝑥∈Ω having the property that, for every 𝑣∈𝑊₀^1,𝑝(𝑥)(Ω), the function 𝑥→𝑤(𝑥)𝑣(𝑥)∈𝐿¹(Ω) and _Ω|∇𝑢|^{𝑝(𝑥)−2}+|∇𝑢|^{2𝑝(𝑥)−2}1+|∇𝑢|^2𝑝(𝑥)(∇𝑢,∇𝑣)_ℝ^𝑁𝑑𝑥=𝜆_Ω𝑤(𝑥)𝑣(𝑥)𝑑𝑥. (14) We know that 𝑊₀^1,𝑝(𝑥)(Ω) is compactly embedded into 𝐶(Ω) (by 𝑁<𝑝⁻<𝑝^∗(𝑥)). So there is a constant 𝑐₀>0 such that |𝑢|_∞≤𝑐₀‖𝑢‖, for all 𝑢∈𝑊₀^1,𝑝(𝑥)(Ω).
Set Φ(𝑢)=∫_Ω(1/𝑝(𝑥))(|∇𝑢|^𝑝(𝑥)+1+|∇𝑢|^2𝑝(𝑥))𝑑𝑥,  Ψ(𝑢)=∫_Ω𝐹(𝑥,𝑢)𝑑𝑥,𝑢∈𝑊₀^1,𝑝(𝑥)(Ω) and 𝜑(𝑢)=Φ(𝑢)−𝜆Ψ(𝑢), for all 𝑢∈𝑊₀^1,𝑝(𝑥)(Ω).
We know that the critical points of 𝜑 are just the weak solutions of (P).
We consider a non-smooth potential function 𝐹∶Ω×ℝ→ℝ such that 𝐹(𝑥,0)=0 a.e. on Ω satisfying the following conditions:
𝐇(𝐣): (𝐡_𝟏)𝐹(⋅,𝑡) is measurable for all 𝑡∈ℝ;(𝐡_𝟐)𝐹(𝑥,⋅) is locally Lipschitz for a.e. 𝑥∈Ω;(𝐡_𝟑) there exist 𝑎∈𝐿^∞(Ω)₊, 𝑐>0 such that |𝑤|≤𝑎(𝑥)+𝑐|𝑡|^{𝛼(𝑥)−1},a.e.𝑥∈Ω,∀𝑡∈ℝ, (15) where 𝑤∈𝜕𝐹(𝑥,𝑡) and 1<𝛼⁻≤𝛼⁺<𝑝⁻;(𝐡_𝟒) there exists 𝑞∈𝐶(Ω) with 𝑝⁺<𝑞⁻≤𝑞(𝑥)<𝑝^∗(𝑥), such that lim_|𝑡|→0(𝐹(𝑥,𝑡)/|𝑡|^𝑞(𝑥))=0 uniformly a.e. 𝑥∈Ω;(𝐡_𝟓)sup_𝑡∈ℝ𝐹(𝑥,𝑡)>0, for all 𝑥∈Ω.

Theorem 6. Let (𝐡_𝟏)--(𝐡_𝟓) hold. Then, there are an open interval Λ⊆[0,+∞) and a number 𝜎 such that, for every 𝜆 belonging to Λ, problem (P) possesses at least three solutions in 𝑊₀^1,𝑝(𝑥)(Ω) whose norms are less than 𝜎.

Proof. We observe that Ψ(𝑢) is Lipschitz on 𝐿^𝛼(𝑥)(Ω) and, taking into account that 𝛼(𝑥)<𝑝^∗(𝑥), Ψ is also locally Lipschitz on 𝑊₀^1,𝑝(𝑥)(Ω) (see Proposition 2.2 of [15]). Moreover it results in 𝜕Ψ(𝑢)⊆∫_Ω𝜕𝐹(𝑥,𝑢)𝑑𝑥 (see [24]). The interpretation of 𝜕Ψ(𝑢)⊆∫_Ω𝜕𝐹(𝑥,𝑢)𝑑𝑥 is as follows: to every 𝑤∈𝜕Ψ(𝑢) there corresponds a mapping 𝑤(𝑥)∈𝜕𝐹(𝑥,𝑢) for almost all 𝑥∈Ω having the property that for every 𝑣∈𝑊₀^1,𝑝(𝑥)(Ω) the function 𝑤(𝑥)𝑣(𝑥)∈𝐿¹(Ω) and ⟨𝑤,𝑣⟩=∫_Ω𝑤(𝑥)𝑣(𝑥)𝑑𝑥 (see [24]). The proof is divided into the following five steps.
Step  1. We show that 𝜑 is coercive.
By (𝐡_𝟐), for almost all 𝑥∈Ω, 𝑡↦𝐹(𝑥,𝑡) is differentiable almost everywhere on ℝ and we have 𝑑𝐹()().𝑑𝑡𝑥,𝑡∈𝜕𝐹𝑥,𝑡 (16) From (𝐡_𝟑), there exist positive constants 𝑎₁,𝑎₂ such that 𝐹(𝑥,𝑡)=𝐹(𝑥,0)+^𝑡₀𝑑𝐹()(𝑥)𝑐𝑑𝑠𝑥,𝑠𝑑𝑠≤𝑎𝑡+𝛼(𝑥)|𝑡|^𝛼(𝑥)≤𝑎₁+𝑎₂|𝑡|^𝛼(𝑥) (17) for a.e. 𝑥∈Ω and 𝑡∈ℝ.
Note that 1<𝛼(𝑥)≤𝛼⁺<𝑝⁻<𝑝^∗(𝑥); then by Proposition 2, we have 𝑊₀^1,𝑝(𝑥)(Ω)↪𝐿^𝛼(𝑥)(Ω) (compact embedding). Furthermore, there exists 𝑐₁ such that |𝑢|_𝛼(𝑥)≤𝑐₁‖𝑢‖.
So, for |𝑢|_𝛼(𝑥)>1 and ‖𝑢‖>1, we have ∫_Ω|𝑢|^𝛼(𝑥)𝑑𝑥≤|𝑢|^𝛼+_𝛼(𝑥)≤𝑐^𝛼+₁‖𝑢‖^𝛼+.
Hence, 𝜑(𝑢)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+1+|∇𝑢|^2𝑝(𝑥)×𝑑𝑥−𝜆_Ω𝐹()≥2𝑥,𝑢𝑑𝑥𝑝⁺_Ω|∇𝑢|^𝑝(𝑥)𝑑𝑥−𝜆_Ω𝐹()≥2𝑥,𝑢𝑑𝑥𝑝⁺‖𝑢‖^𝑝−−𝜆𝑎₁meas(Ω)−𝜆𝑎₂𝑐^𝛼+₁‖𝑢‖^𝛼+→+∞, (18) as ‖𝑢‖→+∞.
Step  2. We show that 𝜑 is weakly lower semicontinuous.
Let 𝑢_𝑛⇀𝑢 weakly in 𝑊₀^1,𝑝(𝑥)(Ω), and by Proposition 2, we obtain the following results: 𝑊₀^1,𝑝(𝑥)(Ω)↪𝐿^𝑝(𝑥)(Ω);𝑢_𝑛→𝑢in𝐿^𝑝(𝑥)(Ω);𝑢_𝑛→𝑢fora.a𝐹.𝑥∈Ω;𝑥,𝑢_𝑛(𝑥)→𝐹(𝑥,𝑢(𝑥))fora.a.𝑥∈Ω. (19) By Fatou's lemma, we have limsup_𝑛→∞Ω𝐹𝑥,𝑢_𝑛(𝑥)𝑑𝑥≤_Ω𝐹(𝑥,𝑢(𝑥))𝑑𝑥. (20) Thus, liminf_𝑛→∞𝜑𝑢_𝑛=_Ω1𝑝(𝑥)∇𝑢_𝑛^𝑝(𝑥)+1+∇𝑢_𝑛^2𝑝(𝑥)𝑑𝑥−𝜆limsup_𝑛→∞Ω𝐹𝑥,𝑢_𝑛≥𝑑𝑥_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+1+|∇𝑢|^2𝑝(𝑥)𝑑𝑥−𝜆_Ω𝐹(𝑥,𝑢)𝑑𝑥=𝜑(𝑢). (21)
Step  3. We show that (PS)-condition holds.
Suppose {𝑢_𝑛}_𝑛≥1⊆𝑊₀^1,𝑝(𝑥)(Ω) such that |𝜑(𝑢_𝑛)|≤𝑐 and 𝑚(𝑢_𝑛)→0 as 𝑛→+∞. If 𝑢^∗_𝑛∈𝜕𝜑(𝑢_𝑛) is such that 𝑚(𝑢_𝑛)=‖𝑢^∗_𝑛‖_(𝑊0^1,𝑝(𝑥)_)∗,𝑛≥1, then we know that 𝑢^∗_𝑛=Φ^′𝑢_𝑛−𝜆𝑤_𝑛, (22) where the nonlinear operator Φ^′∶𝑊₀^1,𝑝(𝑥)→(𝑊₀^1,𝑝(𝑥))^∗ is defined as Φ^′(𝑢)=,𝑣_Ω|∇𝑢|^{𝑝(𝑥)−2}+|∇𝑢|^{2𝑝(𝑥)−2}1+|∇𝑢|^2𝑝(𝑥)(∇𝑢,∇𝑣)_ℝ^𝑁𝑑𝑥, (23) for all 𝑢,𝑣∈𝑊₀^1,𝑝(𝑥)(Ω). From the work of Chang [1], we know that if 𝑤_𝑛∈𝜕Ψ(𝑢_𝑛), then 𝑤_𝑛∈𝐿^{𝛼′(𝑥)}(Ω), where 1/𝛼^′(𝑥)+1/𝛼(𝑥)=1.
Since 𝜑 is coercive, {𝑢_𝑛}_𝑛≥1 is bounded in 𝑊₀^1,𝑝(𝑥)(Ω) and there exists 𝑢∈𝑊₀^1,𝑝(𝑥)(Ω) such that a subsequence of {𝑢_𝑛}_𝑛≥1, which is still denoted as {𝑢_𝑛}_𝑛≥1, satisfies 𝑢_𝑛⇀𝑢 weakly in 𝑊₀^1,𝑝(𝑥)(Ω). Next we will prove that 𝑢_𝑛→𝑢 in 𝑊₀^1,𝑝(𝑥)(Ω).
By 𝑊₀^1,𝑝(𝑥)(Ω)→𝐿^𝛼(𝑥)(Ω), we have 𝑢_𝑛→𝑢 in 𝐿^𝛼(𝑥)(Ω). Moreover, since ‖𝑢^∗_𝑛‖_∗→0, we get |⟨𝑢^∗_𝑛,𝑢_𝑛⟩|≤𝜀_𝑛.
Since 𝑢^∗_𝑛=Φ^′(𝑢_𝑛)−𝜆𝑤_𝑛, we obtain Φ^′𝑢_𝑛,𝑢_𝑛−𝑢−𝜆_Ω𝑤_𝑛𝑢_𝑛−𝑢𝑑𝑥≤𝜀_𝑛,∀𝑛≥1. (24) Moreover, since 𝑢_𝑛→𝑢 in 𝐿^𝛼(𝑥)(Ω) and {𝑤_𝑛}_𝑛≥1 are bounded in 𝐿^{𝛼′(𝑥)}(Ω), where 1/𝛼(𝑥)+1/𝛼^′(𝑥)=1, one has ∫_Ω𝑤_𝑛(𝑢_𝑛−𝑢)𝑑𝑥→0. Therefore, limsup_𝑛→∞Φ^′𝑢_𝑛,𝑢_𝑛−𝑢≤0. (25) But we know that Φ^′ is a mapping of type (𝑆₊) (by Proposition 3). Thus we obtain 𝑢_𝑛→𝑢in𝑊₀^1,𝑝(𝑥)(Ω). (26)
Step  4. There exists a 𝑢₁∈𝑊₀^1,𝑝(𝑥)(Ω)⧵{0} such that Ψ(𝑢₁)>0.
By (𝐡_𝟓), for each 𝑥∈Ω, there is 𝑡_𝑥∈ℝ such that 𝐹(𝑥,𝑡_𝑥)>0.
For 𝑥∈ℝ^𝑁, denote by 𝑁_𝑥 a neighborhood of 𝑥 which is the product of 𝑁 compact intervals. From (𝐡_𝟓) and 𝐹(𝑥,𝑡)∈𝐶(Ω×ℝ), for any 𝑥₀∈Ω, there are 𝑁_𝑥0⊂ℝ^𝑁, 𝑡_𝑥0∈ℝ and 𝛿₀>0, such that 𝐹(𝑥,𝑡_𝑥0)>𝛿₀>0 for all 𝑥∈𝑁_𝑥0⋂Ω.
Since Ω⊆ℝ^𝑁 is bounded, Ω is compact. Then we can find 𝑁_𝑥1,𝑁_𝑥2,…,𝑁_𝑥𝑛 such that Ω⊂⋃^𝑛_𝑖=1𝑁_𝑥𝑖 and 𝑁_𝑥𝑖⋂𝑁_𝑥𝑗=𝜕𝑁_𝑥𝑖⋂𝜕𝑁_𝑥𝑗,(𝑖≠𝑗) and, also, we can find 𝑡_𝑥1,𝑡_𝑥2,…,𝑡_𝑥𝑛∈ℝ, and 𝑛 positive numbers 𝛿₁,𝛿₂,…,𝛿_𝑛 such that 𝐹𝑥,𝑡_𝑥𝑖>𝛿_𝑖>0uniformlyfor𝑥∈𝑁_𝑥𝑖Ω,𝑖=1,2,…,𝑛. (27)
Now, set 𝛿₀=min{𝛿₁,𝛿₂,…,𝛿_𝑛}, and 𝑡₀=max{𝑡_𝑥1,𝑡_𝑥2,…,𝑡_𝑥𝑛}, and sup_{|𝑡|<𝑡0;𝑥∈Ω}|𝐹(𝑥,𝑡)|=𝑀. (28) Then, we can find a closed set Ω_𝑥𝑖⊂int(𝑁_𝑥𝑖⋂Ω) such that measΩ_𝑥𝑖>𝑀meas𝑁_𝑥𝑖⋂Ω𝛿₀,+𝑀 (29) where meas(𝐴) denote the Lebesgue measure of set 𝐴. We consider a function 𝑢₁∈𝑊₀^1,𝑝(𝑥)(Ω) such that |𝑢₁(𝑥)|∈[0,𝑡₀] and 𝑢₁(𝑥)≡𝑡_𝑥𝑖 for all 𝑥∈Ω_𝑥𝑖. For instance, we can set 𝑢₁(𝑥)=∑^𝑛_𝑖=1𝑢^𝑖₁(𝑥), where 𝑢^𝑖₁∈𝐶^∞₀(𝑁_𝑥𝑖⋂Ω) and 𝑢^𝑖₁(𝑥)=𝑡_𝑥𝑖,𝑥∈Ω_𝑥𝑖,0≤𝑢^𝑖₁(𝑥)<𝑡_𝑥𝑖,𝑥∈𝑁_𝑥𝑖∩Ω⧵Ω_𝑥𝑖. (30) Then, from (27)–(29), we have Ψ𝑢₁=_Ω𝐹𝑥,𝑢₁𝑑𝑥=_{⋃𝑛𝑖=1𝑁𝑥𝑖∩Ω}𝐹𝑥,𝑢₁=𝑑𝑥_{⋃𝑛𝑖=1Ω𝑥𝑖}𝐹𝑥,𝑢₁+𝑑𝑥_{(⋃𝑛𝑖=1𝑁𝑥𝑖⋃∩Ω)⧵𝑛𝑖=1Ω𝑥𝑖}𝐹𝑥,𝑢₁≥𝑑𝑥_𝑛𝑖=1𝛿_𝑖measΩ_𝑥𝑖−^𝑛_𝑖=1𝑀meas𝑁_𝑥𝑖Ω−measΩ_𝑥𝑖>^𝑛_𝑖=1𝛿₀+𝑀measΩ_𝑥𝑖−𝑀meas𝑁_𝑥𝑖Ω>0. (31)
Step  5. We show that Φ, Ψ satisfy conditions (1) and (2) of Theorem 4.
Let 𝑢₀=0; then we can easily find Φ(𝑢₀)=Ψ(𝑢₀)=0.
From (7) and Proposition 1, we have the following:
if ‖𝑢‖≥1, then 2𝑝⁺‖𝑢‖^𝑝−(𝑢)≤|Ω|≤Φ2+𝑝⁻‖𝑢‖^𝑝+; (32) if ‖𝑢‖<1, then 2𝑝⁺‖𝑢‖^𝑝+(𝑢)≤|Ω|≤Φ2+𝑝⁻. (33) From (𝐡_𝟒), there exist 𝜂∈]0,1[ and 𝐶₃>0 such that 𝐹(𝑥,𝑡)≤𝐶₃|𝑡|^𝑞(𝑥)≤𝐶₃|𝑡|^𝑞−,∀𝑡∈−𝜂,𝜂,𝑥∈Ω. (34) In view of (𝐡_𝟑), if we put 𝐶₄=max𝐶₃,sup_{|𝑡|𝜂≤<1}𝑎₁+𝑎₂|𝑡|^𝛼−|𝑡|^𝑞−,sup_|𝑡|≥1𝑎₁+𝑎₂|𝑡|^𝛼+|𝑡|^𝑞−, (35) then we have 𝐹(𝑥,𝑡)≤𝐶₄|𝑡|^𝑞−,∀𝑡∈ℝ,𝑥∈Ω. (36) Fix 𝑟 such that 0<𝑟<1. And when (2/𝑝⁺)max{‖𝑢‖^𝑝−,‖𝑢‖^𝑝+}<𝑟<1, by Sobolev Embedding Theorem (𝑊₀^1,𝑝(𝑥)(Ω)↪𝐿^𝑞−(Ω)), we have (for suitable positive constants 𝐶₅,𝐶₆) Ψ(𝑢)=_Ω𝐹(𝑥,𝑢)𝑑𝑥≤𝐶_4Ω|𝑢|^𝑞−𝑑𝑥≤𝐶₅‖𝑢‖^𝑞−<𝐶₆𝑟^{𝑞−/𝑝−}or𝐶₆𝑟^{𝑞−/𝑝+}. (37) Since 𝑞⁻>𝑝⁺≥𝑝⁻, we have lim_𝑟→0+sup_{(2/𝑝+)max{‖𝑢‖𝑝}⁻_{,‖𝑢‖𝑝}⁺_}<𝑟Ψ(𝑢)𝑟=0. (38) And so, taking into account (32) and (33), lim_𝑟→0+sup_{Φ(𝑢)<𝑟}Ψ(𝑢)𝑟=0. (39) From Step 4, there exists 𝑢₁∈𝑊₀^1,𝑝(𝑥)(Ω)⧵{0} such that Ψ(𝑢₁)>0. Thanks to (32) and (33), we have 20<𝑝⁺max𝑢₁^𝑝−,𝑢₁^𝑝+≤Φ𝑢₁, (40) and so Ψ𝑢₁Φ𝑢₁>0. (41) By (32), (33), and (39), there exists 𝑟₀<(2/𝑝⁺)max{‖𝑢₁‖^𝑝−,‖𝑢₁‖^𝑝+}≤Φ(𝑢₁) such that, for each 𝑟∈]0,𝑟₀[, sup_{Φ(𝑢)<𝑟}Ψ(𝑢)Ψ𝑢<𝑟¹Φ𝑢₁. (42) By choosing 𝑟∈]0,𝑟₀[, conditions (1) and (2) requested in Theorem 4 are verified and so the proof is complete.

Acknowledgments

This paper is supported by the Fundamental Research Funds for the Central Universities (no. DL12BC10; 2013), the New Century Higher Education Teaching Reform Project of Heilongjiang Province in 2012 (no. JG2012010012), the National Science Foundation of China (nos. 11126286 and 11201095), China Postdoctoral Science Foundation Funded Project (no. 20110491032), and China Postdoctoral Science (Special) Foundation (no. 2012T50303).