Multiple Solutions for Nonhomogeneous Neumann Differential Inclusion Problems by the 𝑝(𝑥)-Laplacian

Abstract

A class of nonlinear Neumann problems driven by 𝑝(𝑥)-Laplacian with a nonsmooth locally Lipschitz potential (hemivariational inequality) was considered. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, Weierstrass theorem and Mountain Pass theorem are used to prove the existence of at least two nontrivial solutions.

1. Introduction

Recently, there are several papers on the research of the Neumann-type problems involving the 𝑝(𝑥)-Laplacian. Of the existing works in the literature, the majority deal with problems in which the potential function is smooth (i.e., 𝐹(𝑥,⋅)∈𝐶¹(ℝ)). We mention the works of Mihailescu [1], Fan and Ji [2], Yao [3], Shi and Ding [4] and Cammaroto et al. [5]. Problems with a nonsmooth potential, were studied by Dai [6, 7], who for the case 𝑁<𝑝⁻<+∞ established the existence of three or infinitely many solutions for Neumann-type differential inclusion problems involving the 𝑝(𝑥)-Laplacian, using the nonsmooth three-critical-points theorem and nonsmooth Ricceri type variational principle, respectively. Not long ago, Qian et al. [8] studied the nonhomogeneous Neumann problem with indefinite weight; that is, −div|∇𝑢|^{𝑝(𝑥)−2}∇𝑢+𝑉(𝑥)|𝑢|^{𝑝(𝑥)−2}𝑢∈𝜕𝐹(𝑥,𝑢(𝑥)),in||Ω,∇𝑢^{𝑝(𝑥)−2}𝜕𝑢𝜕𝑛∈𝜕𝐺𝑥,𝛾₀(𝑢(𝑥)),on𝜕Ω, (1) where Ω⊂ℝ^𝑁 is a bounded domain with smooth boundary 𝜕Ω, √2𝑝⁻>𝑁, 𝑉∈𝐿^∞(Ω) is a function possibly changing sign, 𝛾₀∶𝑊^1,𝑝(𝑥)(Ω)→𝐿^𝑝(𝑥)(𝜕Ω) is the trace operator with 𝛾₀(𝑢)=𝑢|_𝜕Ω for all 𝑢∈𝑊^1,𝑝(𝑥)(Ω), 𝐹(𝑥,𝑡) and 𝐺(𝑥,𝑡) are locally Lipschitz functions in the 𝑡-variable integrand (in general it can be nonsmooth), 𝜕𝐹(𝑥,𝑡) and 𝜕𝐺(𝑥,𝑡) are the subdifferentials with respect to the 𝑡-variable in the sense of Clarke [9]. The authors prove the existence of at least one nontrivial solution of (1) using the nonsmooth Mountain Pass theorem and Weierstrass theorem.

If 𝑉(𝑥)≡1, then problem (1) becomes problem (2) as follows: −div|∇𝑢|^{𝑝(𝑥)−2}∇𝑢+|𝑢|^{𝑝(𝑥)−2}𝑢∈𝜕𝐹(𝑥,𝑢),in||Ω,∇𝑢^{𝑝(𝑥)−2}𝜕𝑢()𝜕𝑛∈𝜕𝐺𝑥,𝑢,on𝜕Ω. (2) In this paper, our goal is to establish the existence of at least two nontrivial solutions for problem (2).

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 has not the first eigenvalue. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years. These problems are interesting in applications to modeling electrorheological fluids (see [10, 11]) and image restoration (see [12]).

This paper is divided into three sections: in the second section, we introduce some necessary knowledge on the nonsmooth analysis and 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 potentials 𝐹(𝑥,𝑡),  𝐺(𝑥,𝑡) and prove the multiplicity results for problem (2).

2. Preliminary

In this section, we first review some facts on variable exponent spaces 𝐿^𝑝(𝑥)(Ω) and 𝑊^1,𝑝(𝑥)(Ω). For the details, see [13–18].

Firstly, we need to give some notations, which we will use through this paper: 𝐶₊Ω=𝑝∈𝐶Ω∶𝑝(𝑥)>1forany𝑥∈Ω,𝑝⁻=min_𝑥∈Ω𝑝(𝑥),𝑝⁺=max_𝑥∈Ω𝑝(𝑥)forany𝑝∈𝐶₊Ω. (3)

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), define 𝐿^𝑝(𝑥)(Ω)=𝑢∈𝒰(Ω)∶_Ω|𝑢|^𝑝(𝑥)𝑑𝑥<+∞ (4) with the norm |𝑢|_𝐿^𝑝(𝑥)_(Ω)=|𝑢|_𝑝(𝑥)=inf𝜆>0∶_Ω𝑢𝜆^𝑝(𝑥),𝑊𝑑𝑥≤1_1,𝑝(𝑥)(Ω)=𝑢∈𝐿^𝑝(𝑥)(Ω)∶|∇𝑢|∈𝐿^𝑝(𝑥)(Ω) (5) with the norm ‖𝑢‖=‖𝑢‖_𝑊^1,𝑝(𝑥)_(Ω)=|𝑢|_𝑝(𝑥)+|∇𝑢|_𝑝(𝑥). (6)

Denote 𝑝^∗(𝑥)=𝑁𝑝(𝑥)(𝑥)(𝑥)(𝑥)𝑝𝑁−𝑝,𝑝<𝑁,+∞,𝑝≥𝑁,_∗(𝑥)=(𝑁−1)𝑝(𝑥)(𝑥)(𝑥)(𝑥)𝑁−𝑝,𝑝<𝑁,+∞,𝑝≥𝑁. (7)

Let 𝑋 be a Banach space and 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‖, for all 𝑦,𝑧∈𝑈.

For a locally Lipschitz function 𝜑∶𝑋↦ℝ, we define 𝜑⁰(𝑥;ℎ)=limsup_{𝑥′→𝑥;𝜆↓0}𝜑𝑥^′+𝜆ℎ−𝜑𝑥^′𝜆. (8)

It is obvious that the function ℎ↦𝜑⁰(𝑥;ℎ) is sublinear and continuous and so is the support function of a nonempty, convex, and 𝑤^∗-compact set 𝜕𝜑(𝑥)⊆𝑋^∗, defined by 𝜕𝜑(𝑥)=𝑥^∗∈𝑋^∗;𝑥^∗,ℎ≤𝜑⁰(𝑥;ℎ),∀ℎ∈𝑋. (9)

The multifunction 𝜕𝜑∶𝑋↦2^𝑋∗ is called the generalized subdifferential of 𝜑.

If 𝜑 is also convex, then 𝜕𝜑(𝑥) coincides with subdifferential in the sense of convex analysis, defined by 𝜕_𝐶𝜑(𝑥)=𝑥^∗∈𝑋^∗∶𝑥^∗,ℎ≤𝜑(𝑥+ℎ)−𝜑(𝑥)∀ℎ∈𝑋. (10) If 𝜑∈𝐶¹(𝑋), then 𝜕𝜑(𝑥)={𝜑^′(𝑥)}.

A point 𝑥∈𝑋 is a critical point of 𝜑 if 0∈𝜕𝜑(𝑥). It is easily seen that if 𝑥∈𝑋 is a local minimum of 𝜑, then 0∈𝜕𝜑(𝑥).

A locally Lipschitz function 𝜑∶𝑋↦ℝ satisfies the nonsmooth 𝐶-condition at level 𝑐∈𝑅 (the nonsmooth 𝐶𝑐-condition for short), if for every sequence {𝑥_𝑛}_𝑛≥1⊆𝑋, such that 𝜑(𝑥_𝑛)→𝑐 and (1+‖𝑥_𝑛‖)𝑚(𝑥_𝑛)→0, as 𝑛→+∞, there is a strongly convergent subsequence, where 𝑚(𝑥_𝑛)=inf{‖𝑥^∗‖_∗∶𝑥^∗∈𝜕𝜑(𝑥_𝑛)}. If this condition is satisfied at every level 𝑐∈ℝ, then we say that 𝜑 satisfies the nonsmooth 𝐶-condition.

Lemma 1 (see [19]). Consider the following.(1)The spaces 𝐿^𝑝(𝑥)(Ω) and 𝑊^1,𝑝(𝑥)(Ω) are separable and reflexive Banach spaces. Moreover, 𝐿^𝑝(𝑥)(Ω) is uniform convex.(2)If 𝑞∈𝐶₊(Ω) and 𝑞(𝑥)<𝑝^∗(𝑥) for any 𝑥∈Ω, then the imbedding from 𝑊^1,𝑝(𝑥)(Ω) to 𝐿^𝑞(𝑥)(Ω) is compact and continuous.(3)If 𝑞∈𝐶₊(𝜕Ω) and 𝑞(𝑥)<𝑝_∗(𝑥) for any 𝑥∈𝜕Ω, then the imbedding from 𝑊^1,𝑝(𝑥)(Ω) to 𝐿^𝑞(𝑥)(𝜕Ω) is compact and continuous.

Lemma 2 (see [15]). 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|_𝐿^𝑝(𝑥)_(Ω)|𝑣|_𝐿^𝑞(𝑥)_(Ω). (11)

Lemma 3 (see [15]). Set 𝐼(𝑢)=∫_Ω(|∇𝑢|^𝑝(𝑥)+|𝑢|^𝑝(𝑥))𝑑𝑥. If 𝑢,𝑢_𝑘∈𝑊^1,𝑝(𝑥)(Ω), then(1)for 𝑢≠0,‖𝑢‖=𝜆⇔𝐼(𝑢/𝜆)=1;(2)‖𝑢‖<1(=1;>1)⇔𝐼(𝑢)<1(=1;>1); (3)‖𝑢‖>1⇒‖𝑢‖^𝑝−≤𝐼(𝑢)≤‖𝑢‖^𝑝+;‖𝑢‖<1⇒‖𝑢‖^𝑝+≤𝐼(𝑢)≤‖𝑢‖^𝑝−; (4)lim_𝑘→+∞‖𝑢_𝑘‖=0⇔lim_𝑘→+∞𝐼(𝑢_𝑘)=0;‖𝑢_𝑘‖→+∞⇔𝐼(𝑢_𝑘)→+∞.

In this paper, we denote by 𝑋=∶𝑊^1,𝑝(𝑥)(Ω); 𝑋^∗=∶(𝑊^1,𝑝(𝑥)(Ω))^∗ the dual space and by ⟨⋅,⋅⟩ the dual pair. Consider the following function: 𝐽(𝑢)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+|𝑢|^𝑝(𝑥)𝑑𝑥,𝑢∈𝑋. (12) We know that (see [20]) 𝐽∈𝐶¹(𝑋,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^′∶𝑋→𝑋^∗; then ⟨ℒ(𝑢),𝑣⟩=∫_Ω(|∇𝑢|^{𝑝(𝑥)−2}∇𝑢⋅∇𝑣+|𝑢|^{𝑝(𝑥)−2}𝑢𝑣)𝑑𝑥, for all 𝑢,𝑣∈𝑋.

Lemma 4 (see [19]). Set 𝑋=𝑊^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₊); that is, if 𝑢_𝑛⇀𝑢 weakly in 𝑋 and limsup_𝑛→∞⟨ℒ(𝑢_𝑛),𝑢_𝑛−𝑢⟩≤0, implies that 𝑢_𝑛→𝑢 in 𝑋;(3)ℒ∶𝑋→𝑋^∗ is a homeomorphism.

The following theorem, which is used as a theoretical basis in this paper, is a nonsmooth version of the well-known Mountain Pass theorem (see Chang [19] or Kourogenis and Papageorgiou [21]).

Theorem 5. 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. The Main Result and Proof of the Theorem

In this section, we will discuss the existence of weak solution of (2).

Our hypotheses on nonsmooth potential 𝐹(𝑥,𝑡) and 𝐺(𝑥,𝑡) are given as follows. 𝐻(𝐹)∶𝐹∶Ω×ℝ→ℝ is a function such that 𝐹(𝑥,0)=0 almost everywhere on Ω and satisfies the following facts: (1) for all 𝑡∈ℝ, 𝑥↦𝐹(𝑥,𝑡) is measurable; (2) for almost all 𝑥∈Ω, 𝑡↦𝐹(𝑥,𝑡) is locally Lipschitz. 𝐻(𝐺)∶𝐺∶𝜕Ω×ℝ→ℝ is a function such that 𝐺(𝑥,0)=0 almost everywhere on 𝜕Ω and satisfies the following facts: (1) for all 𝑡∈ℝ, 𝑥↦𝐺(𝑥,𝑡) is measurable; (2) for almost all 𝑥∈𝜕Ω, 𝑡↦𝐺(𝑥,𝑡) is locally Lipschitz.

We consider the energy function 𝜑∶𝑋→ℝ for problem (2), defined by 𝜑(𝑢)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+|𝑢|^𝑝(𝑥)𝑑𝑥−_Ω𝐹(𝑥,𝑢)𝑑𝑥−_𝜕Ω𝐺(𝑥,𝑢)𝑑𝜎,𝑢∈𝑋, (13) where 𝑑𝜎 is the surface measure on 𝜕Ω.

Lemma 6. Suppose that 𝐻(𝐹), 𝐻(𝐺), and the following conditions hold:(𝑓₁)there exist 𝑐₁>0, 𝛼∈𝐶(Ω) with 1<𝛼(𝑥)≤𝛼⁺<𝑝⁻ such that |𝑤|≤𝑐₁|𝑡|^{𝛼(𝑥)−1} (14) for almost all 𝑥∈Ω, all 𝑡∈ℝ and 𝑤∈𝜕𝐹(𝑥,𝑡);(𝑔₁)there exist 𝑐₂>0, 𝛽∈𝐶(𝜕Ω) with 1<𝛽(𝑥)≤𝛽⁺<𝑝⁻<𝑝_∗(𝑥) such that |𝑤|≤𝑐₂|𝑡|^{𝛽(𝑥)−1} (15) for almost all 𝑥∈𝜕Ω, all 𝑡∈ℝ and 𝑤∈𝜕𝐺(𝑥,𝑡).
Then, 𝜑 is locally Lipschitz in 𝑋.

Proof. By 𝐽∈𝐶¹(𝑋,ℝ), we have 𝐽(𝑢₁)−𝐽(𝑢₂)=𝐽^′(𝑢)⋅(𝑢₁−𝑢₂), where 𝑢=𝑡𝑢₁+(1−𝑡)𝑢₂, 𝑡∈(0,1).
Let 𝐵_𝑟={𝑥∈𝑋∶‖𝑢−𝑢₀‖_𝑋≤𝑟}.
Note that 𝐵_𝑟 is 𝑤-compact. Then, we obtain that there exists a positive constant 𝑀, such that ‖𝐽^′(𝑢)‖_𝑋^∗≤𝑀 for sufficiently small 𝑟.
Therefore, for any 𝑢₁,𝑢₂∈𝐵_𝑟, we have 𝐽𝑢₁−𝐽𝑢₂=𝐽(𝑢)⋅𝑢₁−𝑢₂≤𝐽^′(𝑢)_𝑋∗𝑢₁−𝑢_2𝑋≤𝑀𝑢₁−𝑢_2𝑋. (16)
On the other hand, by (𝑓₁) and Lebourg mean value theorem, we have 𝐹𝑥,𝑢₁−𝐹𝑥,𝑢₂≤𝑐₁|𝑢|^{𝛼(𝑥)−1}𝑢₁−𝑢₂. (17)
Hence, _Ω𝐹𝑥,𝑢₁𝑑𝑥−_Ω𝐹𝑥,𝑢₂𝑑𝑥≤𝑐_1Ω|𝑢|^{𝛼(𝑥)−1}𝑢₁−𝑢₂𝑑𝑥≤𝑐₂|𝑢|^{𝛼(𝑥)−1}_{𝐿𝛼′(𝑥)(Ω)}𝑢₁−𝑢_{2𝐿𝛼(𝑥)(Ω)}, (18) where (1/𝛼^′(𝑥))+(1/𝛼(𝑥))=1.
Obviously, it is verified that _Ω|𝑢|^{𝛼(𝑥)−1𝛼′(𝑥)}𝑑𝑥=_Ω|𝑢|^𝛼(𝑥)≤|𝑑𝑥𝑢|^𝛼+_𝛼(𝑥)≤𝑐‖𝑢‖^𝛼+,|𝑢|_𝛼(𝑥)|>1,𝑢|^𝛼−_𝛼(𝑥)≤𝑐‖𝑢‖^𝛼−,|𝑢|_𝛼(𝑥)<1 (19) is bounded, since 𝑢∈𝐵_𝑟.
So, _Ω𝐹𝑥,𝑢₁𝑑𝑥−_Ω𝐹𝑥,𝑢₂𝑑𝑥≤𝑐₃𝑢₁−𝑢_2𝛼(𝑥)≤𝑐𝑢₁−𝑢₂, (20) since 𝑊^1,𝑝(𝑥)(Ω)↪𝐿^𝛼(𝑥)(Ω) is a compact imbedding.
As above, there is a positive constant 𝑐₄, such that _𝜕Ω𝐺𝑥,𝑢₁𝑑𝜎−_𝜕Ω𝐺𝑥,𝑢₂𝑑𝜎≤𝑐₄𝑢₁−𝑢_{2𝐿𝛽(𝑥)(𝜕Ω)}≤𝑐𝑢₁−𝑢₂, (21) since 𝑊^1,𝑝(𝑥)(Ω)↪𝐿^𝛽(𝑥)(𝜕Ω) is a compact imbedding.
Therefore, 𝜑 is locally Lipschitz.

Remark 7. If assumptions (𝑓₁) and (𝑔₁) in Lemma 6 are replaced, respectively, by the following:(𝑓₁)there exist 𝑐₁,𝑐₁>0,𝛼∈𝐶(Ω) with 1<𝛼(𝑥)≤𝛼⁺<𝑝⁻<𝑝^∗(𝑥) such that |𝑤|≤𝑐₁|𝑡|^{𝛼(𝑥)−1}+𝑐₁ (22) for almost all 𝑥∈Ω, all 𝑡∈ℝ and 𝑤∈𝜕𝐹(𝑥,𝑡);(𝑔₁)there exist 𝑐₂,𝑐₂>0, 𝛽∈𝐶(𝜕Ω) with 1<𝛽(𝑥)≤𝛽⁺<𝑝⁻<𝑝_∗(𝑥), such that |𝑤|≤𝑐₂|𝑡|^{𝛽(𝑥)−1}+𝑐₂ (23) for almost all 𝑥∈𝜕Ω, all 𝑡∈ℝ and 𝑤∈𝜕𝐺(𝑥,𝑡), then the result of Lemma 6 is also correct.

Theorem 8. Let 𝐻(𝐹), 𝐻(𝐺), (𝑓₁),(𝑔₁), and the following conditions (𝑓₂)-(𝑓₃),(𝑔₂) hold:(𝑓₂)there exist 𝑐₅>0,𝑥₀∈Ω and 0<𝑟<1, such that 𝑤≥𝑐₅|𝑡−𝛿|^𝛽0−1,∀𝑥∈𝐵_2𝑟𝑥₀,𝛿<𝑡≤1, (24) where 1<𝛽₀<𝑝⁻,𝑤∈𝜕𝐹(𝑥,𝑡);(𝑓₃)𝑤𝑡≤0, for all 𝑥∈Ω,𝑤∈𝜕𝐹(𝑥,𝑡) and 0<|𝑡|≤𝛿;(𝑔₂)𝑣𝑡≤0, for all 𝑥∈𝜕Ω,𝑣∈𝜕𝐺(𝑥,𝑡) and 0<|𝑡|≤𝛿, where 10<𝛿<min2𝑐₅𝛽₀𝑟^𝑝+𝑝⁻𝛼⁻2^𝑁+𝛽0𝛼⁻𝑟^𝑝++2^𝑝++2𝑐₁𝑝⁻𝑟^{𝑝+1/(𝑝−−𝛽0)},12. (25)
Then, 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 this step.
Firstly, for almost all 𝑥∈Ω, by 𝐻(𝐹) (2), 𝑡↦𝐹(𝑥,𝑡) is differentiable almost everywhere on ℝ and we have 𝑑𝐹()().𝑑𝑡𝑥,𝑡∈𝜕𝐹𝑥,𝑡 (26)
Moreover, from (𝑓₁), we have 𝐹(𝑥,𝑡)=𝐹(𝑥,0)+^𝑡₀𝑑𝐹()𝑐𝑑𝑠𝑥,𝑠𝑑𝑠≤¹𝛼(𝑥)|𝑡|^𝛼(𝑥), (27) for almost all 𝑥∈Ω and 𝑡∈ℝ.
Note that 1<𝛼(𝑥)≤𝛼⁺<𝑝⁻<𝑝^∗(𝑥) and 1<𝛽(𝑥)≤𝛽⁺<𝑝⁻<𝑝_∗(𝑥); then by Lemma 1, we have 𝑊^1,𝑝(𝑥)(Ω)↪𝐿^𝛼(𝑥)(Ω) and 𝑊^1,𝑝(𝑥)(Ω)↪𝐿^𝛽(𝑥)(𝜕Ω) (compact imbedding).
Furthermore, there exist 𝑐₆,𝑐₆ such that |𝑢|_𝐿^𝛼(𝑥)_(Ω)≤𝑐₆‖𝑢‖,|𝑢|_𝐿^𝛽(𝑥)_(𝜕Ω)≤𝑐₆‖𝑢‖. (28)
So, for any |𝑢|_𝐿^𝛼(𝑥)_(Ω)>1, |𝑢|_𝐿^𝛽(𝑥)_(𝜕Ω)>1 and ‖𝑢‖>1, we have _Ω|𝑢|^𝛼(𝑥)𝑑𝑥≤|𝑢|^𝛼+_{𝐿𝛼(𝑥)(Ω)}≤𝑐^𝛼+₆‖𝑢‖^𝛼+,_𝜕Ω|𝑢|^𝛽(𝑥)𝑑𝑥≤|𝑢|^𝛽+_{𝐿𝛽(𝑥)()𝜕Ω}≤𝑐^𝛽+₆‖𝑢‖^𝛽+. (29)
Hence, 𝜑(𝑢)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+|𝑢|^𝑝(𝑥)𝑑𝑥−_Ω𝐹(𝑥,𝑢)𝑑𝑥−_𝜕Ω𝐺()≥1𝑥,𝑢𝑑𝜎𝑝⁺‖𝑢‖^𝑝−−𝑐₁𝛼⁻𝑐^𝛼+₆‖𝑢‖^𝛼+−𝑐₂𝛽⁻𝑐^𝛽+₆‖𝑢‖^𝛽+⟶∞,as‖𝑢‖⟶∞. (30)
Step  2. We will show that the 𝜑 is weakly lower semicontinuous.
Let 𝑢_𝑛⇀𝑢 weakly in 𝑊^1,𝑝(𝑥)(Ω); by Lemma 1, we obtain the following results: 𝑊^1,𝑝(𝑥)(Ω)↪𝐿^𝑝(𝑥)(Ω),𝑊^1,𝑝(𝑥)(Ω)↪𝐿^𝑝(𝑥)();𝑢𝜕Ω_𝑛⟶𝑢in𝐿^𝑝(𝑥)(Ω),𝑢_𝑛⟶𝑢in𝐿^𝑝(𝑥)();𝑢𝜕Ω_𝑛(𝑥)⟶𝑢(𝑥)a.a𝑢.𝑥∈Ω,_𝑛𝑦⟶𝑢𝑦a.a𝐹.𝑦∈𝜕Ω;𝑥,𝑢_𝑛(𝑥)⟶𝐹(𝑥,𝑢(𝑥))a.a𝐺.𝑥∈Ω,𝑥,𝑢_𝑛𝑦⟶𝐺𝑥,𝑢𝑦a.a.𝑦∈𝜕Ω. (31)
By Fatou’s lemma, we have limsup_𝑛→∞Ω𝐹𝑥,𝑢_𝑛(𝑥)𝑑𝑥≤_Ω𝐹(𝑥,𝑢(𝑥))𝑑𝑥,limsup_{𝑛→∞𝜕Ω}𝐺𝑥,𝑢_𝑛(𝑥)𝑑𝜎≤_𝜕Ω𝐺(𝑥,𝑢(𝑥))𝑑𝜎. (32)
Thus, liminf_𝑛→∞𝜑𝑢_𝑛=liminf_𝑛→∞𝐽𝑢_𝑛−limsup_𝑛→∞Ω𝐹𝑥,𝑢_𝑛𝑑𝑥−limsup_{𝑛→∞𝜕Ω}𝐺𝑥,𝑢_𝑛(𝑢)−𝑑𝜎≥𝐽_Ω𝐹(𝑥,𝑢)𝑑𝑥−_𝜕Ω𝐺(𝑥,𝑢)𝑑𝜎=𝜑(𝑢). (33)
Hence, by Weierstrass theorem, we deduce that there exists a global minimizer 𝑢₀∈𝑊^1,𝑝(𝑥)(Ω) such that 𝜑𝑢₀=min_{𝑢∈𝑊1,𝑝(𝑥)(Ω)}𝜑(𝑢). (34)
Step  3. In this step, we prove that 𝜑(𝑢₀)<0.
By (𝑓₂), we have 𝐹()≥𝑐𝑥,𝑡⁵𝛽₀|𝑡−𝛿|^𝛽0−𝑐₁𝛼⁻𝛿^𝛼(𝑥),∀𝑥∈𝐵_2𝑟𝑥₀,𝛿<𝑡≤1. (35)
Suppose that 𝑥₀∈Ω and 𝐵_2𝑟(𝑥₀)⊆Ω with 2𝑟<1. Let 𝜂∈𝐶^∞₀(𝐵_2𝑟(𝑥₀)) such that 𝜂=1,𝑥∈𝐵_𝑟(𝑥₀);0≤𝜂(𝑥)≤1 and |∇𝜂|≤2/𝑟. Denote 𝑠=2𝛿; then 𝜑𝑠𝜂=_Ω1𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)+𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)𝑑𝑥−_Ω𝐹𝑥,𝑠𝜂𝑑𝑥−_𝜕Ω𝐺=𝑥,𝑠𝜂𝑑𝜎_{𝐵2𝑟(𝑥0)}1𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)+𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)−𝑑𝑥_{𝐵2𝑟(𝑥0)}𝐹=𝑥,𝑠𝜂𝑑𝑥_{𝐵2𝑟(𝑥0)}1𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)+𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)−𝑑𝑥_{𝐵2𝑟(𝑥0)∩{𝑥∣0<𝜂≤(1/2)}}𝐹𝑥,𝑠𝜂𝑑𝑥+_{𝐵2𝑟(𝑥0)∩{𝑥∣(1/2)<𝜂≤1}}𝐹≤1𝑥,𝑠𝜂𝑑𝑥𝑝⁻2𝑟^𝑝++1_{𝐵2𝑟(𝑥0)}𝑠^𝑝(𝑥)+𝑐𝑑𝑥₁𝛼⁻_{𝐵2𝑟(𝑥0)∩{𝑥∣0<𝜂≤(1/2)}}|𝑠|^𝛼(𝑥)𝜂^𝛼(𝑥)+𝑐𝑑𝑥₁𝛼⁻_{𝐵2𝑟(𝑥0)∩{𝑥∣(1/2)<𝜂(𝑥)≤1}}𝛿^𝛼(𝑥)−𝑑𝑥_{𝐵2𝑟(𝑥0)∩{𝑥∣(1/2)<𝜂≤1}}𝑐₅𝛽₀𝑠^𝛽01𝜂−2^𝛽0≤1𝑑𝑥𝑝⁻2𝑟^𝑝++1𝑠^𝑝−+𝑐₁𝛼⁻𝑠^𝛼−+𝑐₁𝛼⁻𝛿^𝛼−×meas𝐵_2𝑟𝑥₀−_{𝐵𝑟(𝑥0)}𝑐₅𝛽₀𝑠^𝛽012^𝛽0≤1𝑑𝑥𝑝⁻2𝑟^𝑝++1𝑠^𝛼−+𝑐₁𝛼⁻𝑠^𝛼−+𝑐₁𝛼⁻𝛿^𝛼−×meas𝐵_2𝑟𝑥₀−𝑐₅𝛽₀12^𝛽0meas𝐵_𝑟𝑥₀=meas𝐵_𝑟𝑥₀×1𝑝⁻2𝑟^𝑝++1𝑠^𝛼−+𝑐₁𝛼⁻𝑠^𝛼−+𝑐₁𝛼⁻𝛿^𝛼−2^𝑁−𝑐₅𝛽₀12^𝛽0𝑠^𝛽0=meas𝐵_𝑟𝑥₀×1𝑝⁻2𝑟^𝑝++1𝑠^𝛼−+𝑐₁𝛼⁻𝑠^𝛼−+𝑐₁𝛼⁻𝑠^𝛼−2^𝛼−2^𝑁−𝑐₅𝛽₀12^𝛽0𝑠^𝛽0<𝑠^𝛽0meas𝐵_𝑟𝑥₀×1𝑝⁻2𝑟^𝑝++𝑐+1¹𝛼⁻+𝑐₁𝛼⁻𝑠^{𝛼−−𝛽0}2^𝑁−𝑐₅𝛽₀12^𝛽0<0. (36)
Step  4. We will show that there exists another nontrivial weak solution of problem (2).
Let 𝜌∈𝐶(Ω) and 𝑝⁺<𝜌⁻≤𝜌⁺<𝑝_∗(𝑥).
For 𝑢∈𝑋 with ‖𝑢‖<1, by Lemma 1, we have _Ω|𝑢|^𝜌(𝑥)𝑑𝑥≤𝑐₇‖𝑢‖^𝜌−,_𝜕Ω|𝑢|^𝜌(𝑥)𝑑𝑥≤𝑐₇‖𝑢‖^𝜌−, (37) where 𝑐₇ and 𝑐₇ are two positive constants.
By (𝑓₃), we have 0∈𝜕𝐹(𝑥,0),∀𝑥∈Ω. (38)
From Lebourg’s mean value theorem and (𝑓₃), we obtain 𝐹()()∈⟨()⟩=1𝑥,𝑡−𝐹𝑥,0𝜕𝐹𝑥,𝜆𝑡,𝑡𝜆⟨𝜕𝐹(𝑥,𝜆𝑡),𝜆𝑡⟩≤0, (39) where 𝜆∈(0,1),0<|𝑡|≤𝛿.
Thus, 𝐹(𝑥,𝑡)≤𝐹(𝑥,0)=0,∀𝑥∈Ω,0<|𝑡|≤𝛿. (40)
Set 𝑢₀=𝑟₀,𝑟𝑑=min⁰2,12𝑐₇,12𝑐₇,𝛼⁻𝛽⁻2𝑝⁺𝑐₁𝑐₇𝛽⁻+𝑐₂𝑐₇𝛼⁻𝛿^{(𝛼−−𝜌+)1/(𝜌−−𝑝−)}. (41)
Divide Ω into two parts: Ω₁={𝑥∈Ω∶0<|𝑢(𝑥)|≤𝛿} and Ω₂={𝑥∈Ω∶|𝑢(𝑥)|≥𝛿}.
For any 𝑢∈𝑋 such that ‖𝑢‖=𝑑, we have _Ω𝐹(𝑥,𝑢)𝑑𝑥=_Ω1𝐹(𝑥,𝑢)𝑑𝑥+_Ω2𝐹(𝑥,𝑢)𝑑𝑥≤_Ω2𝐹()𝑐𝑥,𝑢𝑑𝑥≤¹𝛼⁻_Ω2|𝑢|^𝛼(𝑥)=𝑐𝑑𝑥₁𝛼⁻_Ω2|𝑢|^𝜌(𝑥)|𝑢|^{𝛼(𝑥)−𝜌(𝑥)}≤𝑐𝑑𝑥₁𝛼⁻𝛿^{(𝛼−−𝜌+)}_Ω2|𝑢|^𝜌(𝑥)≤𝑐𝑑𝑥₁𝛼⁻𝛿^{(𝛼−−𝜌+)}_Ω|𝑢|^𝜌(𝑥)𝑑𝑥. (42) As above, we have _𝜕Ω𝐺()𝑐𝑥,𝑢𝑑𝜎≤²𝛽⁻𝛿^{(𝛽−−𝜌+)}_𝜕Ω|𝑢|^𝜌(𝑥)𝑑𝑥. (43) Hence, 𝜑(𝑢)=_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+|𝑢|^𝑝(𝑥)𝑑𝑥−_Ω𝐹(𝑥,𝑢)𝑑𝑥−_𝜕Ω𝐺()≥1𝑥,𝑢𝑑𝜎𝑝⁺‖𝑢‖^𝑝−−𝑐₁𝛼⁻𝑐₇𝛿^{(𝛼−−𝜌+)}‖𝑢‖^𝜌−−𝑐₂𝛽⁻𝑐₇𝛿^{(𝛼−−𝜌+)}‖𝑢‖^𝜌−≥1𝑝⁺‖𝑢‖^𝑝−𝑐1−¹𝛼⁻𝑐₇+𝑐₂𝛽⁻𝑐₇𝛿^{(𝛼−−𝜌+)}𝑝⁺‖𝑢‖^{𝜌−−𝑝−}≥12𝑝⁺𝑑^𝑝−>0. (44)
Note that 𝜑 is coercive; hence, it satisfies the nonsmooth 𝐶-condition. So by the nonsmooth Mountain Pass theorem (consequence of Theorem 5), there exists a 𝑢₁∈𝑊^1,𝑝(𝑥)(Ω) such that 𝜑𝑢₁=𝑐>0,𝑚𝑢₁=0, (45) which means that 𝑢₁ is another nontrivial critical point of 𝜑.
Using the similar and simpler arguments, we can prove the following theorems.

Theorem 9. Let 𝐻(𝐹), 𝐻(𝐺), (𝑓₁),(𝑓₂),(𝑔₁), and the following conditions (𝑓^′₃),(𝑔^′₂) hold:(𝑓^′₃)𝐹(𝑥,𝑡)=0, for all 𝑥∈Ω,0<|𝑡|≤𝛿;(𝑔^′₂)𝐺(𝑥,𝑡)=0, for  all  𝑥∈𝜕Ω,0<|𝑡|≤𝛿, where 10<𝛿<min2𝑐₅𝑟^𝑝+𝑝⁻𝛽₀2^𝑁+𝛽0𝑟^𝑝++2^{𝑝+1/(𝑝−−𝛽0)},12. (46)
Then, the problem (2) has at least two nontrivial solutions.

Proof. The steps are similar to those of Theorem 8. In fact, we only need to modify Step  3 as follows: (3^′) show that the 𝜑(𝑢₀)<0 under the assumptions of Theorem 9. Then, from Steps  1, 2, 3^′, and 4 above, the problem (2) has at least two nontrivial solutions.
Step 3^′. By (𝑓₂), we have 𝐹()≥𝑐𝑥,𝑡⁵𝛽₀|𝑡−𝛿|^𝛽0,∀𝑥∈𝐵_2𝑟𝑥₀,𝛿<𝑡≤1. (47)
Suppose that 𝑥₀∈Ω and 𝐵_2𝑟(𝑥₀)⊆Ω with 2𝑟<1.
Let 𝜂∈𝐶^∞₀(𝐵_2𝑟(𝑥₀)) such that 𝜂=1,𝑥∈𝐵_𝑟(𝑥₀);0≤𝜂(𝑥)≤1 and |∇𝜂|≤(2/𝑟). Denote 𝑠=2𝛿; then 𝜑𝑠𝜂=_Ω1𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)+𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)𝑑𝑥−_Ω𝐹𝑥,𝑠𝜂𝑑𝑥−_𝜕Ω𝐺=𝑥,𝑠𝜂𝑑𝜎_{𝐵2𝑟(𝑥0)}1𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)+𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)−𝑑𝑥_{𝐵2𝑟(𝑥0)}𝐹=𝑥,𝑠𝜂𝑑𝑥_{𝐵2𝑟(𝑥0)}1𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)+𝜂^𝑝(𝑥)𝑠^𝑝(𝑥)−𝑑𝑥_{𝐵2𝑟(𝑥0)∩{𝑥∣0<𝜂≤(1/2)}}𝐹𝑥,𝑠𝜂𝑑𝑥+_{𝐵2𝑟(𝑥0)∩{𝑥∣(1/2)<𝜂≤1}}𝐹≤1𝑥,𝑠𝜂𝑑𝑥𝑝⁻2𝑟^𝑝++1_{𝐵2𝑟(𝑥0)}𝑠^𝑝(𝑥)−𝑑𝑥_{𝐵2𝑟(𝑥0)∩{𝑥∣(1/2)<𝜂≤1}}𝑐₅𝛽₀𝑠^𝛽01𝜂−2^𝛽0≤1𝑑𝑥𝑝⁻2𝑟^𝑝++1𝑠^𝑝−meas𝐵_2𝑟𝑥₀−_{𝐵𝑟(𝑥0)}𝑐₅𝛽₀𝑠^𝛽012^𝛽0=1𝑑𝑥𝑝⁻2𝑟^𝑝++1𝑠^𝑝−meas𝐵_2𝑟𝑥₀−𝑐₅𝛽₀12^𝛽0meas𝐵_𝑟𝑥₀=meas𝐵_𝑟𝑥₀×1𝑝⁻2𝑟^𝑝++1𝑠^𝑝−2^𝑁−𝑐₅𝛽₀12^𝛽0𝑠^𝛽0=𝑠^𝛽0meas𝐵_𝑟𝑥₀×1𝑝⁻2𝑟^𝑝++1𝑠^{𝑝−−𝛽0}2^𝑁−𝑐₅𝛽₀12^𝛽0<0. (48) This is the end.

Corollary 10. Let 𝐻(𝐹), 𝐻(𝐺), (𝑓₂),  (𝑓₁), and (𝑔₁) hold; then the problem (2) has at least a nontrivial solution.

Proof. From Remark 7 and Steps  1–3, by Weierstrass theorem, the functional 𝜑 has a critical point, which is just the solution of problem (2).

Remark 11. Let 𝑝⁻>max{𝛼⁺,𝛽⁺,𝜃⁺} and consider the following two nonsmooth locally Lipschitz functions: 𝐹()=|𝑡|𝑥,𝑡|0,0≤<𝛿,max|𝑡−𝛿^𝜃(𝑥),|𝑡−𝛿|^𝛼(𝑥)|,𝑡≥𝛿,max|𝑡+𝛿^𝜃(𝑥),|𝑡+𝛿|^𝛼(𝑥)𝐺()=|𝑡||,𝑡≤−𝛿,𝑥,𝑡0,0≤<𝛿,max|𝑡−𝛿^𝜃(𝑥),|𝑡−𝛿|^𝛽(𝑥)|,𝑡≥𝛿,max|𝑡+𝛿^𝜃(𝑥),|𝑡+𝛿|^𝛽(𝑥),𝑡≤−𝛿, (49) where 0<𝛿<1, inf_𝑥∈Ω(𝛼(𝑥)−𝜃(𝑥))>0, inf_{𝑥∈𝜕Ω}(𝛽(𝑥)−𝜃(𝑥))>0 and 𝜃⁻>1. In the following, we will show that 𝐹(𝑥,𝑡) satisfies hypotheses 𝐻(𝐹) and (𝑓₁)−(𝑓₃), and 𝐺(𝑥,𝑡) satisfies hypotheses 𝐻(𝐺) and (𝑔₁)-(𝑔₂).
Note that 𝑡↦|𝑡−𝛿|^𝜃(𝑥) and 𝑡↦|𝑡−𝛿|^𝛼(𝑥) are convex functions; thus, 𝐹(𝑥,𝑡) is also convex. Since 𝑡↦|𝑡−𝛿|^𝜃(𝑥), 𝑡→|𝑡−𝛿|^𝛼(𝑥) are locally Lipschitz, so 𝑡↦𝐹(𝑥,𝑡) is locally Lipschitz. Thus, 𝑡↦𝐹(𝑥,𝑡) is regular. Then ()=|𝑡|𝜃(𝑥)𝜕𝐹𝑥,𝑡0,0≤≤𝛿,)(𝑡−𝛿^{𝜃(𝑥)−1}{(𝑥)+()𝛼(𝑥)[𝛼(𝑥),𝛿<𝑡<1+𝛿,𝜆𝜃1−𝜆∶𝜆∈0,1]},𝑡=1+𝛿,)(𝑡−𝛿^{𝛼(𝑥)−1}(𝑥),𝑡>1+𝛿,−𝜃)(−𝑡−𝛿^{𝜃(𝑥)−1}{(𝑥)−()𝜃(𝑥)[(𝑥),−1−𝛿<𝑡<−𝛿,−𝜆𝛼1−𝜆∶𝜆∈0,1]},𝑡=−1−𝛿,−𝛼)(−𝑡−𝛿^{𝛼(𝑥)−1},𝑡<−1−𝛿. (50)
Hence, for any 𝑤∈𝜕𝐹(𝑥,𝑡), we have |𝑤|≤|𝑡|𝜃0,0≤≤𝛿,₊(𝑡−𝛿)^{𝜃(𝑥)−1}<𝜃⁺<𝜃⁺1𝛿^(𝛼+−1)|𝑡|^{𝛼(𝑥)−1}𝜃,𝛿<𝑡<1+𝛿,₊(−𝑡−𝛿)^{𝜃(𝑥)−1}<𝜃⁺<𝜃⁺1𝛿^(𝛼+−1)|𝑥|^{𝛼(𝑥)−1}𝛼,−1−𝛿<𝑡<−𝛿₊+𝜃⁺<𝛼⁺+𝜃⁺|𝑡|^{𝛼(𝑥)−1},|𝑡|𝛼=1+𝛿,₊|𝑡|^{𝛼(𝑥)−1},|𝑡|>1+𝛿. (51)
Thus, we have |𝑤|≤𝛼⁺+𝜃⁺+1𝛿^(𝛼+−1)|𝑡|^{𝛼(𝑥)−1}(),(𝑥),∀𝑤∈𝜕𝐹𝑥,𝑡𝑤=𝜃)(𝑡−𝛿^{𝜃(𝑥)−1}≥𝜃⁻(𝑡−𝛿)^𝜃+−1,∀𝑡∈(𝛿,1]. (52)
Therefore, conditions 𝐻(𝐹) and (𝑓₁)−(𝑓₃) hold. In a similar fashion, we have that conditions 𝐻(𝐺) and (𝑔₁)-(𝑔₂) hold.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (nos. DL12BC10 and 2014), the New Century Higher Education Teaching Reform Project of Heilongjiang Province in 2012 (no. JG2012010012), Humanities and Social Science Project in Heilongjiang Province Department of Education: Empirical Analysis and Construction of the Evaluation on Complex Resources service performance of Hybrid Library Based on Comprehensive Weight, the National Science Found of China (nos. 11126286 and 11201095), China Postdoctoral Science Foundation Funded Project (no. 20110491032), and China Postdoctoral Science (Special) Foundation (no. 2012T50303).