Existence Results for a 𝑝(𝑥)-Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition

Abstract

We consider the existence and multiplicity of solutions for the 𝑝(𝑥)-Kirchhoff-type equations without Ambrosetti-Rabinowitz condition. Using the Mountain Pass Lemma, the Fountain Theorem, and its dual, the existence of solutions and infinitely many solutions were obtained, respectively.

1. Introduction

The Kirchhoff equation 𝜌𝜕²𝑢𝜕𝑡²−𝜌₀ℎ+𝐸2𝐿^𝐿₀𝜕𝑢𝜕𝜕𝑥𝑑𝑥²𝑢𝜕𝑥²=0 (1) was introduced by Kirchhoff [1] in the study of oscillations of stretched strings and plates, where 𝜌,𝜌₀,ℎ,𝐸, and𝐿 are constants. The stationary analogue of the Kirchhoff equation, that is, (1), is as follows: −𝑎+𝑏_Ω|∇𝑢|²𝑑𝑥Δ𝑢=𝑓(𝑥,𝑢). (2) After the excellent work of Lions [2], problem (2) has received more attention; see [3–10] and references therein.

The 𝑝(𝑥)-Laplace operator arises from various phenomena, for instance, the image restoration [11], the electro-rheological fluids [12], and the thermoconvective flows of non-Newtonian fluids [13, 14]. The study of the 𝑝(𝑥)-Laplace operator is based on the theory of the generalized Lebesgue space 𝐿^𝑝(𝑥)(Ω) and the Sobolev space 𝑊^{𝑚,𝑝(𝑥)}(Ω), which we called variable exponent Lebesgue and Sobolev space. We refer the reader to [15–19] for an overview on the variable exponent Sobo-lev space, and to [20–29] for the study of the 𝑝(𝑥)-Laplacian-type equations.

Recently, there has been an increasing interest in studying the Kirchhoff equation involving the 𝑝(𝑥)-Laplace operator. Autuori et al. [30, 31] have dealt with the nonstationary Kirchhoff-type equation involving the 𝑝(𝑥)-Laplacian of the form 𝑢_𝑡𝑡−𝑀_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥Δ_𝑝(𝑥)𝑢+𝑄𝑡,𝑥,𝑢,𝑢_𝑡()𝑢+𝑓𝑥,𝑢=0,_𝑡𝑡−𝑀_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥Δ_𝑝(𝑥)𝑢+𝜇|∇𝑢|^{𝑝(𝑥)−2}𝑢+𝑄𝑡,𝑥,𝑢,𝑢_𝑡=𝑓(𝑡,𝑥,𝑢). (3)

In [32–35], applying variational method and Ambrosetti-Rabinowitz (AR) condition, Guowei Dai has studied the existence and multiplicity of solutions for the 𝑝(𝑥)-Kirchhoff-type equations with Dirichlet or Neumann boundary condition. In [36], by using (𝑆₊) mapping theory and the Mountain Pass Lemma, Fan has discussed the nonlocal 𝑝(𝑥)-Laplacian Dirichlet problem with the nonvariational form −𝐴(𝑢)Δ_𝑝(𝑥)𝑢=𝐵(𝑢)𝑓(𝑥,𝑢),inΩ,𝑢=0,on𝜕Ω, (4) and the 𝑝(𝑥)-Kirchhoff-type equation with the variational form −𝑎_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥Δ_𝑝(𝑥)𝑢=𝑏_Ω𝐹(𝑥,𝑢)𝑑𝑥𝑓(𝑥,𝑢),inΩ,𝑢=0,on𝜕Ω, (5) under (AR) condition, where 𝐴,𝐵 are two functionals defined on 𝑊₀^1,𝑝(𝑥)(Ω), and 𝐹(𝑥,𝑡)=∫^𝑡₀𝑓(𝑥,𝑠)𝑑𝑠.

Motivated by the above works, the purpose of this paper is to study the 𝑝(𝑥)-Kirchhoff-type equation −𝑎+𝑏_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥Δ_𝑝(𝑥)𝑢=𝑓(𝑥,𝑢),inΩ,𝑢=0,on𝜕Ω, (6) without (AR) condition, where Ω is a smooth bounded domain in ℝ^𝑁, 𝑎,𝑏 are two positive constants, Δ_𝑝(𝑥)𝑢=div(|∇𝑢(𝑥)|^{𝑝(𝑥)−2}∇𝑢(𝑥)), 𝑝∈𝐶^0,𝛽(Ω) for some 𝛽∈(0,1), and 1<𝑝⁻∶=inf_Ω𝑝(𝑥)≤𝑝⁺∶=sup_Ω𝑝(𝑥)<+∞. (7) By taking the famous Mountain Pass Lemma, the Fountain Theorem, and its dual, we obtain the existence of solutions and infinitely many solutions for the 𝑝(𝑥)-Kirchhoff-type equation (6) under no (AR) condition.

2. Preliminary

We recall in this section some definitions and properties of variable exponent Lebesgue-Sobolev space. The variable exponent Lebesgue space 𝐿^𝑝(𝑥)(Ω) is defined by 𝐿^𝑝(𝑥)(Ω)=𝑢∶𝑢∶Ω→ℝismeasurable,_Ω|𝑢|^𝑝(𝑥)𝑑𝑥<∞ (8) with the norm |𝑢|_𝐿^𝑝(𝑥)=|𝑢|_𝑝(𝑥)=inf𝜎>0∶_Ω𝑢𝜎^𝑝(𝑥)𝑑𝑥≤1. (9) The variable exponent Sobolev space 𝑊^1,𝑝(𝑥)(Ω) is defined by 𝑊^1,𝑝(𝑥)(Ω)=𝑢∈𝐿^𝑝(𝑥)(Ω)∶|∇𝑢|∈𝐿^𝑝(𝑥)(Ω) (10) with the norm ‖𝑢‖_𝑊^1,𝑝(𝑥)=‖𝑢‖_1,𝑝(𝑥)=|𝑢|_𝑝(𝑥)+|∇𝑢|_𝑝(𝑥). (11) Denote by 𝑊₀^1,𝑝(𝑥)(Ω) the closure of 𝐶^∞₀(Ω) in 𝑊^1,𝑝(𝑥)(Ω). |∇𝑢|_𝑝(𝑥) is an equivalent norm on 𝑊₀^1,𝑝(𝑥)(Ω). In this paper we use the notation ‖𝑢‖=|∇𝑢|_𝑝(𝑥) for 𝑢∈𝑊₀^1,𝑝(𝑥)(Ω). Define 𝑝^∗(𝑥)=𝑁𝑝(𝑥)𝑁−𝑝(𝑥)if𝑝(𝑥)<𝑁,+∞if𝑝(𝑥)≥𝑁. (12) We refer the reader to [36–38] for the elementary properties of the space 𝑊^1,𝑝(𝑥)(Ω).

Proposition 1 (see [38]). Set 𝜌(𝑢)=∫_Ω|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥. For any 𝑢∈𝐿^𝑝(𝑥)(Ω), then the following are given:(1)|𝑢|_𝑝(𝑥)=𝜆⇔𝜌(𝑢/𝜆)=1  if  𝑢≠0; (2)|𝑢|_𝑝(𝑥)<1(=1;>1)⇔𝜌(𝑢)<1(=1;>1);(3)|𝑢|^𝑝−_𝑝(𝑥)≤𝜌(𝑢)≤|𝑢|^𝑝+_𝑝(𝑥)  if  |𝑢|_𝑝(𝑥)>1; (4)|𝑢|^𝑝+_𝑝(𝑥)≤𝜌(𝑢)≤|𝑢|^𝑝−_𝑝(𝑥)  if    |𝑢|_𝑝(𝑥)<1; (5)lim_𝑘→+∞|𝑢_𝑘|_𝑝(𝑥)=0⇔lim_𝑘→+∞𝜌(𝑢_𝑘)=0;(6)lim_𝑘→+∞|𝑢_𝑘|_𝑝(𝑥)=+∞⇔lim_𝑘→+∞𝜌(𝑢_𝑘)=+∞.

3. Positive Energy Solution

In this section we discuss the existence of weak solutions of (6). For simplicity we write 𝑋=𝑊₀^1,𝑝(𝑥)(Ω).

First, we state the assumptions on 𝑓 as follows.  (𝑓₀) Let 𝑓∶Ω×ℝ→ℝ be a continuous function, and there exist positive constants 𝑐₁,𝑐₂ such that 𝑓(𝑥,𝑡)≤𝑐₁+𝑐₂|𝑡|^{𝛼(𝑥)−1}, (13) where 𝛼∈𝐶(Ω) and 1<𝛼(𝑥)<𝑝^∗(𝑥) for all 𝑥∈Ω. (𝑓^′₀) Let 𝑓∶Ω×ℝ→ℝ be a continuous function, and there exist positive constants 𝑐₁,𝑐₂ such that 𝑓(𝑥,𝑡)≤𝑐₁+𝑐₂|𝑡|^{𝛼(𝑥)−1}, (14) where 𝛼∈𝐶(Ω) and 𝑝⁺<𝛼(𝑥)<𝑝^∗(𝑥) for all 𝑥∈Ω; 𝑡𝑓(𝑥,𝑡)≥0 for all 𝑡>0. (𝑓₁) Let lim_𝑡→+∞(𝐹(𝑥,𝑡)/|𝑡|^2𝑝+)=+∞, uniformly for 𝑥∈Ω, where 𝐹(𝑥,𝑡)=∫^𝑡₀𝑓(𝑥,𝑠)𝑑𝑠. (𝑓₂) There exists 𝜃≥1 such that 𝜃𝐺(𝑥,𝑡)≥𝐺(𝑥,𝑠𝑡) for (𝑥,𝑡)∈Ω×ℝ and 𝑠∈[0,1], where 𝐺(𝑥,𝑡)=𝑡𝑓(𝑥,𝑡)−2𝑝⁺𝐹(𝑥,𝑡). (15) (𝑓₃) Let lim_𝑡→0(𝐹(𝑥,𝑡)/|𝑡|^𝑝+)=0, uniformly on 𝑥∈Ω. (𝑓^′₃) There exists 𝛿>0, such that 𝐹(𝑥,𝑡)≤0 for 𝑥∈Ω, |𝑡|<𝛿. (𝑓₄) Let 𝑓(𝑥,𝑡)=−𝑓(𝑥,−𝑡) for 𝑥∈Ω and 𝑡∈ℝ. (𝑓₅) Let lim_𝑡→0(𝐹(𝑥,𝑡)/|𝑡|^𝑞+)=0, uniformly on 𝑥∈Ω, where 𝑞∈𝐶(Ω) satisfies 1<𝑞(𝑥)<𝑝(𝑥) for 𝑥∈Ω.

Remark 2. Condition (𝑓₂) was first introduced by Jeanjean [39] for the case 𝑝(𝑥)=2. Let 𝑓(𝑥,𝑡)=2𝑝⁺|𝑡|^2𝑝+−2𝑡ln|𝑡|, then 𝐹(𝑥,𝑡)=|𝑡|^2𝑝+|𝑡|−1ln2𝑝⁺|𝑡|^2𝑝+,𝐺(𝑥,𝑡)=|𝑡|^2𝑝+. (16) It is easy to see that the function 𝑓 does not satisfy (AR) condition, but it satisfies (𝑓₁)–(𝑓₅) and (𝑓^′₃).
Define 𝐼(𝑢)=𝐽(𝑢)−Φ(𝑢), where 𝐽(𝑢)=𝑏𝑎+2_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)Φ(𝑢)=𝑑𝑥,_Ω𝐹(𝑥,𝑢)𝑑𝑢. (17) Then 𝐼∈𝐶¹(𝑋,ℝ).

Proposition 3 (see [38]). Assume that (𝑓₀) hold, then the functional 𝐽∶𝑋→ℝ is sequentially weakly lower semicontinuous, Φ∶𝑋→ℝ is sequentially weakly continuous, and 𝐼 is sequentially weakly lower semicontinuous.

Proposition 4 (see [37]). Assume that (𝑓₀) hold, and let 𝑢₀∈𝑊₀^1,𝑝(𝑥)(Ω) be a local minimizer (resp., a strictly local minimizer) of 𝐼 in the 𝐶¹(Ω) topology. Then 𝑢₀ is a local minimizer (resp., a strictly local minimizer) of 𝐼 in the 𝑊₀^1,𝑝(𝑥)(Ω) topology.

Definition 5. We say that 𝑢∈𝑋 is a weak solution of (6), if 𝑎+𝑏_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥_Ω|∇𝑢|^{𝑝(𝑥)−2}=∇𝑢∇𝑣𝑑𝑥_Ω𝑓(𝑥,𝑢)𝑣𝑑𝑥 (18) for any 𝑣∈𝑋.

Definition 6. Let 𝑋 be a Banach space and 𝐼∈𝐶¹(𝑋,ℝ). Given 𝑐∈ℝ. we say that 𝐼 satisfies the Cerami 𝑐 condition (we denote by(𝐶)_𝑐 the condition), if (i)any bounded sequence {𝑢_𝑛}⊂𝑋 such that 𝐼(𝑢_𝑛)→𝑐 and 𝐼^′(𝑢_𝑛)→0 has a convergent subsequence; (ii)there exist constants 𝛿,𝑅,𝛽>0 such that ‖𝑢‖𝐼^′(𝑢)≥𝛽,∀𝑢∈𝐼⁻¹[𝑐−𝛿,𝑐+𝛿],‖𝑢‖≥𝑅. (19)

If 𝐼∈𝐶¹(𝑋,ℝ) satisfies (𝐶)_𝑐 condition for every 𝑐∈ℝ, then we say that 𝐼 satisfies (𝐶) condition.

Remark 7. Although (PS) condition is stronger than (𝐶) condition, the Deformation Theorem is still valid under (𝐶) condition; we see that the Mountain Pass Lemma, the Fountain Theorem, and its dual are true under (𝐶) condition.

Lemma 8. Assume that conditions (𝑓₀)–(𝑓₂) hold. Then 𝐼 satisfies (𝐶) condition.

Proof. From [36, Proposition 3.1], 𝐼 satisfies (i) of (𝐶) condition.
Now we check that 𝐼 satisfies (ii) of (𝐶) condition. Arguing by contradiction, we may assume that, for some 𝑐∈ℝ, 𝐼𝑢_𝑛⟶𝑐,𝑢_𝑛⟶∞,𝑢_𝑛𝐼^′𝑢_𝑛⟶0. (20) Then we have lim_𝑛→∞𝑎_Ω1𝑝(𝑥)−12𝑝⁺|∇𝑢|^𝑝(𝑥)+𝑏𝑑𝑥2_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)×𝑑𝑥_Ω1𝑝(𝑥)−1𝑝⁺|∇𝑢|^𝑝(𝑥)+1𝑑𝑥2𝑝⁺_Ω𝐺(𝑥,𝑢)𝑑𝑥=lim_𝑛→∞𝐼𝑢_𝑛−12𝑝⁺𝐼^′𝑢_𝑛,𝑢_𝑛=𝑐. (21) Let 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖, then up to a subsequence we may assume that 𝑣_𝑛⇀𝑣in𝑣𝑋,_𝑛⟶𝑣in𝐿^𝛼(𝑥)(Ω),𝑣_𝑛(𝑥)⟶𝑣(𝑥)a.e.𝑥∈Ω. (22) If 𝑣=0, inspired by [13, 14], then we define 𝐼𝑡_𝑛𝑢_𝑛=max_[]𝑡∈0,1𝐼𝑡𝑢_𝑛. (23) For any 𝑚>1/2𝑝⁺, let 𝑤_𝑛=(2𝑚𝑝⁺)^1/𝑝−𝑣_𝑛. Since 𝑤_𝑛→0 in 𝐿^𝛼(𝑥)(Ω) and |𝐹(𝑥,𝑡)|≤𝑐₅+𝑐₆|𝑡|^𝛼(𝑥), (24) by the continuity of 𝐹(𝑥,⋅), 𝐹(⋅,𝑤_𝑛)→0 in 𝐿¹(Ω), thus, lim_𝑛→0Ω𝐹⋅,𝑤_𝑛𝑑𝑥=0. (25) Then for 𝑛 large enough, (2𝑚𝑝⁺)^1/𝑝−/‖𝑢_𝑛‖∈(0,1) and 𝐼𝑡_𝑛𝑢_𝑛≥𝐼𝑤_𝑛=𝑎_Ω1𝑝(𝑥)∇𝑤_𝑛^𝑝(𝑥)+𝑏𝑑𝑥2_Ω1𝑝(𝑥)∇𝑤_𝑛^𝑝(𝑥)𝑑𝑥²−_Ω𝐹𝑥,𝑤_𝑛𝑑𝑥=𝑎_Ω1𝑝(𝑥)2𝑚𝑝^+1/𝑝−∇𝑣_𝑛^𝑝(𝑥)+𝑏𝑑𝑥2_Ω1𝑝(𝑥)2𝑚𝑝^+1/𝑝−∇𝑣_𝑛^𝑝(𝑥)𝑑𝑥²−_Ω𝐹𝑥,𝑤_𝑛≥𝑑𝑥2𝑚𝑎𝑝⁺_Ω∇𝑣_𝑛^𝑝(𝑥)+𝑑𝑥2𝑚²𝑏𝑝^+2Ω∇𝑣_𝑛^𝑝(𝑥)𝑑𝑥²−_Ω𝐹𝑥,𝑤_𝑛≥𝑑𝑥2𝑚𝑎𝑝⁺+2𝑚²𝑏𝑝⁺²−_Ω𝐹𝑥,𝑤_𝑛𝑑𝑥. (26) That is, 𝐼(𝑡_𝑛𝑢_𝑛)→∞. From 𝐼(0)=0 and 𝐼(𝑢_𝑛)→𝑐, we know that 𝑡_𝑛∈(0,1) and 𝑎_Ω∇𝑡_𝑛𝑢_𝑛^𝑝(𝑥)𝑑𝑥+𝑏_Ω1𝑝(𝑥)∇𝑡_𝑛𝑢_𝑛^𝑝(𝑥)𝑑𝑥_Ω∇𝑡_𝑛𝑢_𝑛^𝑝(𝑥)𝑑𝑥−_Ω𝑓𝑥,𝑡_𝑛𝑢_𝑛𝑢_𝑛=𝐼𝑑𝑥_′𝑡_𝑛𝑢_𝑛,𝑡_𝑛𝑢_𝑛=𝑡_𝑛𝑑𝑑𝑡_{𝑡=𝑡𝑛}𝐼𝑡𝑢_𝑛=0. (27) Therefore, from (𝑓₂), we have 𝑎_Ω1𝑝(𝑥)−12𝑝⁺∇𝑢_𝑛^𝑝(𝑥)+𝑏𝑑𝑥2_Ω1𝑝(𝑥)∇𝑢_𝑛^𝑝(𝑥)×𝑑𝑥_Ω1𝑝(𝑥)−1𝑝⁺∇𝑢_𝑛^𝑝(𝑥)+1𝑑𝑥2𝑝⁺_Ω𝐺𝑥,𝑢_𝑛𝑑𝑥≥𝑎_Ω1𝑝(𝑥)−12𝑝⁺∇𝑢_𝑛^𝑝(𝑥)+𝑏𝑑𝑥2_Ω1𝑝(𝑥)∇𝑢_𝑛^𝑝(𝑥)×𝑑𝑥_Ω1𝑝(𝑥)−1𝑝⁺∇𝑢_𝑛^𝑝(𝑥)+1𝑑𝑥2𝑝⁺_Ω𝐺𝑥,𝑡_𝑛𝑢_𝑛𝜃≥𝑎𝑑𝑥𝜃_Ω1𝑝(𝑥)−12𝑝⁺𝑡_𝑛^𝑝(𝑥)∇𝑢_𝑛^𝑝(𝑥)+𝑏𝑑𝑥2𝜃_Ω1𝑝(𝑥)𝑡_𝑛^𝑝(𝑥)∇𝑢_𝑛^𝑝(𝑥)×𝑑𝑥_Ω1𝑝(𝑥)−1𝑝⁺𝑡_𝑛^𝑝(𝑥)∇𝑢_𝑛^𝑝(𝑥)+1𝑑𝑥2𝑝⁺_Ω𝐺𝑥,𝑡_𝑛𝑢_𝑛𝜃=1𝑑𝑥𝜃𝐼𝑡_𝑛𝑢_𝑛−12𝑝⁺𝐼^′𝑡_𝑛𝑢_𝑛,𝑡_𝑛𝑢_𝑛=1𝜃𝐼𝑡_𝑛𝑢_𝑛→∞. (28) This contradicts (21).
If 𝑣≠0, from (20), when ‖𝑢_𝑛‖≥1, 𝑎𝑝⁻𝑢_𝑛^𝑝++𝑏2𝑝⁻²𝑢_𝑛^2𝑝+−(𝑐+𝑜(1))≥_Ω𝐹𝑥,𝑢_𝑛𝑑𝑥. (29) Then from (𝑓₁) we have 𝑎𝑝⁻1𝑢_𝑛^𝑝++𝑏2𝑝⁻²−𝑐+𝑜(1)𝑢_𝑛^2𝑝+≥_Ω𝐹𝑥,𝑢_𝑛𝑢_𝑛^2𝑝+=𝑑𝑥_𝑣𝑛≠0+_𝑣𝑛=0𝐹𝑥,𝑢_𝑛𝑢_𝑛^2𝑝+𝑣_𝑛^2𝑝+=𝑑𝑥_𝑣𝑛≠0𝐹𝑥,𝑢_𝑛𝑢_𝑛^2𝑝+𝑣_𝑛^2𝑝+𝑑𝑥. (30) For 𝑥∈Θ∶={𝑥∈Ω∶𝑣(𝑥)≠0}, |𝑢_𝑛(𝑥)|→+∞. By (𝑓₁) we have 𝐹𝑥,𝑢_𝑛𝑢_𝑛^𝑝+𝑣_𝑛^𝑝+⟶+∞. (31) Note that the Lebesgue measure of Θ is positive; using the Fatou Lemma, we have _𝑣𝑛≠0𝐹𝑥,𝑢_𝑛𝑢_𝑛^2𝑝+𝑣_𝑛^2𝑝+𝑑𝑥⟶+∞. (32) This contradicts (30).
The technique used in this lemma was first introduced by [39, 40].

Theorem 9. Assume that conditions (𝑓₀)–(𝑓₂) and (𝑓₃) (or (𝑓^′₃)) hold. Then (6) has a nontrivial solution with positive energy.

Proof. From Lemma 8, 𝐼 satisfies (𝐶) condition. Let us show that the functional 𝐼 has a Mountain-Pass-type geometry.
Note that 𝐼(0)=0. By (𝑓₃), there exists 𝛿>0,  and  for  any 𝑢∈𝑋 with |𝑢|_𝐿^∞_(Ω)<𝛿, 𝐼(𝑢)=𝑎_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+𝑏𝑑𝑥2_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥²−_Ω𝐹()≥𝑎𝑥,𝑢𝑑𝑥𝑝⁺‖𝑢‖^𝑝++𝑏𝑝⁺²‖𝑢‖^2𝑝+−_Ω𝐹(𝑥,𝑢)𝑑𝑥>0. (33) This shows that 0 is a strictly local minimizer of 𝐼 in the 𝐶(Ω) topology, and hence 0 is a strictly local minimizer of 𝐼 in the 𝐶¹(Ω) topology. By [37, Theorem 1.1], 0 is a strictly local minimizer of 𝐼 in the 𝑊₀^1,𝑝(𝑥)(Ω) topology. Thus there exists 𝑟>0 such that 𝐼(𝑢)>0 for every 𝑢∈𝑋⧵{0} with ‖𝑢‖≤𝑟.
We claim that inf_{‖𝑢‖=𝑟}𝐼(𝑢)>0. To prove this claim, arguing by contradiction, assume that there exists a sequence {𝑢_𝑛}⊂𝑋 with ‖𝑢_𝑛‖=𝑟 such that 𝐼(𝑢_𝑛)→0 as 𝑛→∞. We may assume that 𝑢_𝑛⇀𝑢₀ in 𝑋. Since 𝐼 is sequentially weakly lower semicontinuous, we have that 𝐼(𝑢₀)=0, and hence 𝑢₀=0. Since Φ is sequentially weakly continuous, then we have that Φ(𝑢_𝑛)→Φ(0)=0, and hence 𝐽(𝑢_𝑛)=𝐼(𝑢_𝑛)+Φ(𝑢_𝑛)→0. It follows from this that 𝑢_𝑛→0 in 𝑋 which contradicts with ‖𝑢_𝑛‖=𝑟.
Let 𝑦∈𝑋 with 𝑦>0 in Ω and ‖𝑦‖=1. By (𝑓₀) and (𝑓₁), for 𝑠≥1 we have 𝐼𝑠𝑦=𝑎_Ω1𝑝(𝑥)∇𝑠𝑦^𝑝(𝑥)𝑑𝑥+𝑏_Ω1𝑝(𝑥)∇𝑠𝑦^𝑝(𝑥)𝑑𝑥²−_Ω𝐹≤𝑎𝑥,𝑠𝑦𝑑𝑥𝑝⁻𝑠^𝑝++𝑏𝑝⁻²𝑠^2𝑝+−𝑐₁𝑠^2𝑝+_Ω𝑦^2𝑝+𝑑𝑥+𝑐₂⟶−∞as𝑠⟶+∞. (34) We set 𝑒=𝑠𝑦. Then for 𝑠 large, we obtain ‖𝑒‖>𝑟,𝐼(𝑒)<0. (35) Hence by the famous Mountain Pass Lemma, problem (6) has a nontrivial weak solution with positive energy.

4. Infinitely Many Solutions

Since 𝑋 is a reflexive and separable Banach space, then there exists {𝑒_𝑗}⊂𝑋 and {𝑒^∗_𝑗}⊂𝑋^∗ such that 𝑋=span𝑒_𝑗∶𝑗=1,2,…,𝑋^∗=span𝑒^∗_𝑗,𝑒∶𝑗=1,2,…_𝑖,𝑒^∗_𝑗,=1,𝑖=𝑗,0,𝑖≠𝑗. (36) For convenience, we write 𝑋_𝑗=span{𝑒_𝑗}, 𝑌_𝑘=⊕^𝑘_𝑗=1𝑋_𝑗, 𝑍_𝑘=⊕^∞_𝑗=𝑘𝑋_𝑗.

Lemma 10 (see [21]). If 𝛼∈𝐶(Ω),  1<𝛼(𝑥)<𝑝^∗ for any 𝑥∈Ω, denote 𝛽_𝑘=sup|𝑢|_𝛼(𝑥)∶‖𝑢‖=1,𝑢∈𝑍_𝑘. (37) Then lim_𝑘→+∞𝛽_𝑘=0.

Proposition 11 (Fountain Theorem). Assume that 𝐼∈𝐶¹(𝑋,ℝ) is an even functional. If, for any 𝑘∈ℕ, there exists 𝜌_𝑘>𝑟_𝑘>0 such that (𝐴₁)𝑎_𝑘=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝐼(𝑢)≤0, (𝐴₂)𝑏_𝑘=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝑟𝑘}𝐼(𝑢)→+∞   as  𝑘→∞, (𝐴₃)𝐼  satisfies  (𝐶)_𝑐  condition for every 𝑐>0, then  𝐼  has an unbounded sequence of critical values.

Proposition 12 (Dual Fountain Theorem). Assume that 𝐼∈𝐶¹(𝑋,ℝ) is an even functional. If, for any 𝑘≥𝑘₀, there exists 𝜌_𝑘>𝑟_𝑘>0 such that (𝐵₁)𝑎_𝑘=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝜌𝑘}𝐼(𝑢)≥0,  (𝐵₂)𝑏_𝑘=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝑟𝑘}𝐼(𝑢)<0, (𝐵₃)𝑑_𝑘=inf_{𝑢∈𝑍𝑘,‖𝑢‖≤𝜌𝑘}𝐼(𝑢)→0  as  𝑘→∞, (𝐵₄)𝐼  satisfies  (𝑐)^∗_𝑐  condition for every  𝑐∈[𝑑_𝑘0,0],  then  𝐼  has a sequence of negative critical values converging to 0.

Theorem 13. Assume that the conditions (𝑓^′₀), (𝑓₁)–(𝑓₄) hold. Then (6) has infinitely many solutions {𝑢_𝑘} such that 𝐼(𝑢_𝑘)→∞ as 𝑘→∞.

Proof. By conditions (𝑓^′₀), (𝑓₁), and (𝑓₃), for any 𝜀>0, there exists 𝐶_𝜀 such that 𝐹(𝑥,𝑢)≥𝐶_𝜀|𝑢|^2𝑝+−𝜀|𝑢|^𝑝+,∀(𝑥,𝑢)∈Ω×ℝ. (38) For 𝑢∈𝑌_𝑘, when ‖𝑢‖>1, 𝐼(𝑢)=𝑎_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+𝑏𝑑𝑥2_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥²−_Ω𝐹()≤𝑎𝑥,𝑢𝑑𝑥𝑝⁻‖𝑢‖^𝑝++𝑏2𝑝⁻²‖𝑢‖^2𝑝+−𝐶_𝜀|𝑢|^2𝑝+_2𝑝++𝜀|𝑢|^𝑝+_𝑝+⟶−∞as‖𝑢‖⟶+∞. (39) Then for some 𝜌_𝑘>0 large enough, 𝑎_𝑘∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝐼(𝑢)≤0. (40) On the other hand, by (𝑓^′₀) and (𝑓₃), there exists 𝐶_𝜀>0 such that |𝐹(𝑥,𝑢)|≤𝜀|𝑢|^𝑝++𝐶_𝜀|𝑢|^𝛼(𝑥),∀(𝑥,𝑢)∈Ω×ℝ. (41) Let 𝛽_𝑘∶=sup_{𝑢∈𝑍𝑘,‖𝑢‖=𝜌𝑘}|𝑢|_𝛼−. From Lemma 10,  𝛽_𝑘→0 as 𝑘→∞. For 𝑢∈𝑍_𝑘, when ‖𝑢‖≤1 and 𝜀 small enough, 𝐼(𝑢)=𝑎_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+𝑏𝑑𝑥2_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥²−_Ω𝐹()≥𝑎𝑥,𝑢𝑑𝑥𝑝⁺‖𝑢‖^𝑝−+𝑏2𝑝⁺²‖𝑢‖^2𝑝−−𝐶_𝜀|𝑢|^𝛼−_𝛼−−𝜀|𝑢|^𝑝+_𝑝+≥𝑎2𝑝⁺‖𝑢‖^𝑝−−𝑐|𝑢|^𝛼−_𝛼−≥𝑎2𝑝⁺‖𝑢‖^𝑝−−𝑐𝛽_𝑘‖𝑢‖^𝛼−. (42) If we choose 𝑟_𝑘∶=(𝑎/4𝑐𝑝⁺𝛽^𝛼_𝑘⁻)^{1/(𝛼−−𝑝−)}→∞ as 𝑘→∞, then, for 𝑢∈𝑍_𝑘 with ‖𝑢‖=𝑟_𝑘, 𝐼(𝑢)≥𝑎4𝑝⁺𝑎4𝑐𝑝⁺𝛽^𝛼−_𝑘^{𝑝−/(𝛼−−𝑝−)}∶=𝑏_𝑘, (43) which implies that 𝑏_𝑘∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝑟𝑘}𝐼(𝑢)≥𝑏_𝑘→+∞ as 𝑘→+∞.

Theorem 14. Assume that conditions (𝑓^′₀), (𝑓₁), (𝑓₂), (𝑓₄), and (𝑓₅) hold. Then (6) has infinitely many solutions {𝑢_𝑘} such that 𝐼(𝑢_𝑘)→0 as 𝑘→∞.

Proof. By conditions (𝑓^′₀), (𝑓₁), and (𝑓₅), for any 𝜀>0, there exists 𝐶_𝜀 such that 𝐹(𝑥,𝑢)≥𝐶_𝜀|𝑢|^2𝑝+−𝜀|𝑢|^𝑞+,∀(𝑥,𝑢)∈Ω×ℝ. (44) For 𝑢∈𝑌_𝑘, when ‖𝑢‖ is large enough, 𝐼(𝑢)=𝑎_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+𝑏𝑑𝑥2_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥²−_Ω𝐹()≤𝑎𝑥,𝑢𝑑𝑥𝑝⁻‖𝑢‖^𝑝++𝑏𝑝⁻²‖𝑢‖^2𝑝+−𝐶_𝜀|𝑢|^2𝑝+_2𝑝++𝜀|𝑢|^𝑞−_𝑞⁻⟶−∞as‖𝑢‖⟶+∞. (45) Then for some 𝑟_𝑘>0 large enough, 𝑏_𝑘∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝑟𝑘}𝐼(𝑢)<0. (46) On the other hand, by (𝑓₅), there exists 𝐶_𝜀>0 such that |𝐹(𝑥,𝑢)|≤𝜀|𝑢|^𝑞−+𝐶_𝜀|𝑢|^𝛼(𝑥),∀(𝑥,𝑢)∈Ω×ℝ. (47) Let 𝛽_𝑘∶=sup_{𝑢∈𝑍𝑘,‖𝑢‖=𝑟𝑘}|𝑢|_𝑞−, then 𝛽_𝑘→0 as 𝑘→∞. For 𝑢∈𝑍_𝑘, when ‖𝑢‖ and 𝜀 small enough, 𝐼(𝑢)=𝑎_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)+𝑏𝑑𝑥2_Ω1𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥²−_Ω𝐹()≥𝑎𝑥,𝑢𝑑𝑥𝑝⁺‖𝑢‖^𝑝+−𝑐𝐶_𝜀|𝑢|^𝛼−_𝛼−−𝑐𝜀|𝑢|^𝑞+_𝑞+≥𝑎2𝑝⁺‖𝑢‖^𝑝+−𝑐|𝑢|^𝑞+_𝑞+≥𝑎2𝑝⁺‖𝑢‖^𝑝+−𝑐𝛽^𝑞+_𝑘‖𝑢‖^𝑞+. (48) If we choose 𝜌_𝑘∶=(4𝑐𝑝⁺𝛽^𝑞+_𝑘/𝑎)^{1/(𝑝+−𝑞+)}→0 as 𝑘→∞, then, for 𝑢∈𝑍_𝑘 with ‖𝑢‖=𝜌_𝑘, 𝐼(𝑢)≥𝑐𝛽^𝑞+_𝑘4𝑐𝑝⁺𝛽^𝑞+_𝑘𝑎^{𝑞+/(𝑝+−𝑞+)}∶=𝑎_𝑘, (49) which implies that 𝑎_𝑘∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝜌𝑘}𝐼(𝑢)≥𝑎_𝑘→0 as 𝑘→+∞.
Furthermore, if 𝑢∈𝑍_𝑘 with ‖𝑢‖≤𝜌_𝑘, then 𝐼(𝑢)≥−𝑐𝛽^𝑞−_𝑘𝜌^𝑞−_𝑘⟶0as𝑘⟶∞, (50) which implies that 𝑑_𝑘=inf_{𝑢∈𝑍𝑘,‖𝑢‖≤𝜌𝑘}𝐼(𝑢)→0 as 𝑘→∞.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (11126339, 11201008).