Multiplicity of Solutions for Perturbed Nonhomogeneous Neumann Problem through Orlicz-Sobolev Spaces

Abstract

We investigate the existence of multiple solutions for a class of nonhomogeneous Neumann problem with a perturbed term. By using variational methods and three critical point theorems of B. Ricceri, we establish some new sufficient conditions under which such a problem possesses three solutions in an appropriate Orlicz-Sobolev space.

1. Introduction

Consider the following nonhomogeneous Neumann problem with a perturbed term:−div(𝛼(|∇𝑢|)∇𝑢)+𝛼(|𝑢|)𝑢=𝜆𝑓(𝑥,𝑢)+𝜇𝑔(𝑥,𝑢),inΩ,𝜕𝑢𝜕𝜈=0,on𝜕Ω, 𝑃_𝜆,𝜇 where Ω is a bounded domain in ℝ^𝑁(𝑁≥3) with smooth boundary 𝜕Ω, 𝜈 is the outer normal to 𝜕Ω, 𝑓,𝑔∶Ω×ℝ→ℝ are two Carathéodory functions, 𝜆>0,𝜇≥0 are two parameters, and the function 𝛼∶(0,∞)→ℝ is such that 𝜑(𝑡)∶ℝ→ℝ defined by 𝜑(𝑡)=𝛼(|𝑡|)𝑡,𝑡≠0,0,𝑡=0, (1.1) is an odd, strictly increasing homeomorphism from ℝ to ℝ.

It is well known that these kinds of problems are important in applications in many fields, such as elasticity, fluid dynamics, and image processing (see [1–4]). Since the operator in the divergence form is nonhomogeneous, we introduce Orlicz-Sobolev space which is an appropriate setting for these problems. Such space originated with Nakano [5] and was developed by Musielak and Orlicz [6]. Many properties of Sobolev spaces have been extended to Orlicz-Sobolev space (see [7–10]). Several authors have widely studied the existence of solutions for the relevant problem by means of variational techniques, monotone operator methods, fixed point, and degree theory (see [11–15]). To the best of our knowledge, for the perturbed nonhomogeneous Neumann problem, there has so far been few papers concerning its multiple solutions. Motivated by the above facts, in this paper, we establish some new sufficient conditions under which such a problem possesses three weak solutions in Orlicz-Sobolev space.

This paper is organized as follows. In Section 2, some preliminaries are presented. In Section 3, we discuss the existence of three weak solutions for problem 𝑃_𝜆,𝜇.

2. Preliminaries

We start by recalling some basic facts about Orlicz-Sobolev space. Let 𝜑 be as in Introduction and Φ(𝑡)∶ℝ→ℝ, Φ(𝑡)=^𝑡₀𝜑(𝑠)𝑑𝑠. (2.1) We observe that Φ is, a Young function, that is Φ(0)=0, Φ is convex and lim_𝑡→∞Φ(𝑡)=+∞. Furthermore, since Φ(𝑡)=0 if and only if 𝑡=0, lim_𝑡→0(Φ(𝑡)/𝑡)=0, and lim_𝑡→∞(Φ(𝑡)/𝑡)=+∞, then Φ is called an 𝑁-function. The function Φ^∗ is called the complementary function of Φ and it satisfies Φ^∗(𝑡)=sup{𝑠𝑡−Φ(𝑠);𝑠≥0},∀𝑡≥0. (2.2) Assume that Φ satisfies the following structural hypotheses  (Φ₁) 1<liminf_𝑡→∞(𝑡𝜑(𝑡)/Φ(𝑡))≤𝑝⁰∶=sup_𝑡>0(𝑡𝜑(𝑡)/Φ(𝑡))<∞;  (Φ₂) 𝑁<𝑝₀∶=inf_𝑡>0(𝑡𝜑(𝑡)/Φ(𝑡))<liminf_𝑡→∞(log(Φ(𝑡))/log(𝑡)).  Further, we also assume that the function  (Φ₃) [0,∞)∋𝑡→Φ(√𝑡) is convex.

The Orlicz space 𝐿_Φ(Ω) defined by Φ is the space of measurable functions 𝑢∶Ω→ℝ such that ‖𝑢‖_𝐿Φ∶=sup_Ω𝑢(𝑥)𝑣(𝑥)𝑑𝑥;_ΩΦ^∗(|𝑣(𝑥)|)𝑑𝑥≤1<∞. (2.3) Then (𝐿_Φ(Ω),‖⋅‖_𝐿Φ) is a Banach space whose norm is equivalent to the Luxemburg norm ‖𝑢‖_Φ∶=inf𝑘>0;_ΩΦ|𝑢(𝑥)|𝑘𝑑𝑥≤1. (2.4) We denote by 𝑊¹𝐿_Φ(Ω) the Orlicz-Sobolev space, defined by 𝑊¹𝐿_Φ(Ω)=𝑢∈𝐿_Φ;𝜕𝑢𝜕𝑥_𝑖∈𝐿_Φ,𝑖=1,2,…,𝑁. (2.5) This is a Banach space with respect to the norm ‖𝑢‖_1,Φ=‖|∇𝑢|‖_Φ+‖𝑢‖_Φ. (2.6)

Lemma 2.1 (see [13]). On 𝑊¹𝐿_Φ(Ω) the norms ‖𝑢‖_1,Φ=‖|∇𝑢|‖_Φ+‖𝑢‖_Φ,‖𝑢‖_2,Φ=max‖|∇𝑢|‖_Φ,‖𝑢‖_Φ,‖𝑢‖∶=inf𝜇>0;_ΩΦ|𝑢(𝑥)|𝜇|(𝑥+Φ∇𝑢)|𝜇𝑑𝑥≤1, (2.7) are equivalent. Moreover, for every 𝑢∈𝑊¹𝐿_Φ(Ω), one has ‖𝑢‖≤2‖𝑢‖_2,Φ≤2‖𝑢‖_1,Φ≤4‖𝑢‖. (2.8)

Lemma 2.2. Let 𝑢∈𝑊¹𝐿_Φ(Ω), then ‖𝑢‖^𝑝0≤_Ω[Φ(|𝑢(𝑥)|)+Φ(|∇𝑢(𝑥)|)]𝑑𝑥≤‖𝑢‖^𝑝0‖𝑢‖‖𝑢‖,if>1,_𝑝0≤_Ω[Φ(|𝑢(𝑥)|)+Φ(|∇𝑢(𝑥)|)]𝑑𝑥≤‖𝑢‖^𝑝0,if‖𝑢‖<1. (2.9)

Proof. For the proof of ‖𝑢‖^𝑝0≤_Ω[Φ𝑢(𝑥(𝑥‖𝑢‖‖𝑢‖(|)|)+Φ(|∇𝑢)|)]𝑑𝑥,if>1,_𝑝0≤_Ω[Φ(|𝑢(𝑥)|)+Φ(|∇𝑢(𝑥)|)]𝑑𝑥,if‖𝑢‖<1, (2.10) we can see Lemma 2.2 of the paper [13]. Since 𝑝⁰≥(𝑡𝜑(𝑡))/Φ(𝑡) for all 𝑡≥0, it follows that letting 𝜎>1, we have log(Φ(𝜎𝑡))−log(Φ(𝑡))=_𝑡^𝜎𝑡𝜑(𝑠)Φ(𝑠)𝑑𝑠≤_𝑡^𝜎𝑡𝑝⁰𝑠𝑑𝑠=log𝜎^𝑝0. (2.11) Thus, one has Φ(𝜎𝑡)≤𝜎^𝑝0Φ(𝑡),𝑡>0,𝜎>1. (2.12) Moreover, by the definition of the norm, we remark that _ΩΦ|𝑢(𝑥)|‖𝑢‖|(𝑥+Φ∇𝑢)|‖𝑢‖𝑑𝑥≤1. (2.13) Therefore, we have _Ω[Φ(|𝑢(𝑥)|)+Φ(|∇𝑢(𝑥)|)]𝑑𝑥=_ΩΦ‖𝑢‖|𝑢(𝑥)|‖𝑢‖‖𝑢‖|(𝑥+Φ∇𝑢)|‖𝑢‖≤‖𝑢‖𝑑𝑥_𝑝0ΩΦ|𝑢(𝑥)|‖𝑢‖|(𝑥+Φ∇𝑢)|‖𝑢‖≤‖𝑢‖𝑑𝑥_𝑝0, (2.14) for all ‖𝑢‖>1.
Similar techniques as those used in the proof of (2.12), we have Φ(𝑡)≤𝜏^𝑝0Φ𝑡𝜏,𝑡>0,0<𝜏<1. (2.15) Therefore, we can obtain _Ω[Φ(|𝑢(𝑥)|)+Φ(|∇𝑢(𝑥)|)]𝑑𝑥≤‖𝑢‖^𝑝0_ΩΦ|𝑢(𝑥)|‖𝑢‖|(𝑥+Φ∇𝑢)|‖𝑢‖≤‖𝑢‖𝑑𝑥_𝑝0, (2.16) for all ‖𝑢‖<1.

Lemma 2.3 (see [13]). Let 𝑢∈𝑊¹𝐿_Φ(Ω) and assume that _Ω[Φ(|𝑢(𝑥)|)+Φ(|∇𝑢(𝑥)|)]𝑑𝑥≤𝑟 (2.17) for some 0<𝑟<1, then one has ‖𝑢‖<1.

Lemma 2.4 (see [13]). If 𝑝₀>𝑁, then 𝑊¹𝐿_Φ(Ω) is compactly embedded in 𝐶⁰(Ω) and there exists a constant 𝑐>0 such that ‖𝑢‖_∞≤𝑐‖𝑢‖_1,Φ,∀𝑢∈𝑊¹𝐿_Φ(Ω), (2.18) where ‖𝑢‖_∞∶=sup_𝑥∈Ω|𝑢(𝑥)|.

Now, one recall, a three critical theorem of B. Ricceri. If 𝑋 is a real Banach space, denote by 𝒲_𝑋 (see [16]) the class of all functionals Φ∶𝑋→ℝ possessing the following property: if {𝑢_𝑛} is a sequence in 𝑋 converging weakly to 𝑢 and liminf_𝑛→∞Φ(𝑢_𝑛)≤Φ(𝑢), then {𝑢_𝑛} has a subsequence converging strongly to 𝑢. For example, if 𝑋 is uniformly convex and 𝑔∶[0,+∞)→ℝ is a continuous, strictly increasing function, then, by a classical results, the functional 𝑢→𝑔(‖𝑢‖) belongs to the class 𝒲_𝑋.

Lemma 2.5 (see [16]). Let 𝑋 be a separable and reflexive real Banach space; let 𝐼∶𝑋→ℝ be a coercive, sequentially weakly lower semicontinuous 𝐶¹ functional, belonging to 𝒲_𝑋, bounded on each bounded subset of 𝑋 and whose derivative admits a continuous inverse on 𝑋^∗; 𝐽∶𝑋→ℝ a 𝐶¹ functional with compact derivative. Assume that 𝐼 has a strict local minimum 𝑢₀ with 𝐼(𝑢₀)=𝐽(𝑢₀)=0. Finally, setting 𝛼^′=max0,limsup_{‖𝑢‖→+∞}𝐽(𝑢)𝐼(𝑢),limsup_𝑢→𝑢0𝐽(𝑢)𝐼(𝑢),𝛽^′=sup_{𝑢∈𝐼−1(0,+∞)}𝐽(𝑢)𝐼(𝑢), (2.19) assume that 𝛼^′<𝛽^′. Then for each compact interval [𝑎,𝑏]⊂(1/𝛽^′,1/𝛼^′) (with the conventions (1/0)=+∞,(1/+∞)=0), there exists 𝐵>0 with the following property: for every 𝜆∈[𝑎,𝑏] and every 𝐶¹ functional Ψ∶𝑋→ℝ with compact derivative, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], the equation 𝐼^′(𝑥)=𝜆𝐽^′(𝑥)+𝜇Ψ^′(𝑥) (2.20) has at least three solutions in 𝑋 whose norms are less than 𝐵.

Lemma 2.6 (see [17]). Let 𝑋 be a reflexive real Banach space; 𝑆⊂ℝ an interval, let 𝐼∶𝑋→ℝ be a sequentially weakly lower semicontinuous 𝐶¹ functional, bounded on each bounded subset of 𝑋 and whose derivative admits a continuous inverse on 𝑋^∗; 𝐽∶𝑋→ℝ a 𝐶¹ functional with compact derivative. Assume that lim_{‖𝑢‖→∞}[𝐼(𝑢)−𝜆𝐽(𝑢)]=+∞ (2.21) for all 𝜆∈𝑆, and that there exists 𝜌∈ℝ such that sup_𝜆∈𝑆inf_𝑢∈𝑋𝐼(𝑢)+𝜆𝜌−𝐽(𝑢)<inf_𝑢∈𝑋sup_𝜆∈𝑆𝐼(𝑢)+𝜆𝜌−𝐽(𝑢). (2.22) Then there exist a nonempty open set 𝐴⊂𝑆 and a positive number 𝐵, with the following property: for every 𝜆∈𝐴 and every 𝐶¹ functional Ψ∶𝑋→ℝ with compact derivative, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], the equation 𝐼^′(𝑥)=𝜆𝐽^′(𝑥)+𝜇Ψ^′(𝑥) (2.23) has at least three solutions in 𝑋 whose norms are less than 𝐵.

Lemma 2.7 (see [18]). Let 𝑋 be a nonempty set and 𝐼,𝐽 two real functions on 𝑋. Assume that there are 𝑟>0 and 𝑥₀,𝑥₁∈𝑋 such that 𝐼𝑥₀=𝐽𝑥₀=0,𝐼𝑥₁>𝑟,sup_{𝑥∈𝐼−1()]−∞,𝑟]}𝐽(𝑥)𝐽𝑥<𝑟¹𝐼𝑥₁. (2.24) Then for each 𝜌 satisfying sup_{𝑥∈𝐼−1()]−∞,𝑟]}𝐽(𝑥)𝐽𝑥<𝜌<𝑟¹𝐼𝑥₁, (2.25) one has sup_𝜆≥0inf_𝑢∈𝑋𝐼(𝑢)+𝜆𝜌−𝐽(𝑢)<inf_𝑢∈𝑋sup_𝜆≥0𝐼(𝑢)+𝜆𝜌−𝐽(𝑢). (2.26)

3. Proof of the Main Results

Set 𝛾=inf{(∫_Ω(Φ(|∇𝑢(𝑥)|)+Φ(|𝑢(𝑥)|))𝑑𝑥/∫_Ω𝐹(𝑥,𝑢(𝑥))𝑑𝑥)∶𝑢∈𝑋,∫_Ω𝐹(𝑥,𝑢(𝑥))𝑑𝑥>0}.

Theorem 3.1. Let Φ be a function satisfying the structural hypotheses (Φ₁–Φ₃) and the following conditions hold(𝐻₁)max{limsup_𝜉→0(sup_𝑥∈Ω𝐹(𝑥,𝜉)/|𝜉|^𝑝0),limsup_|𝜉|→∞(sup_𝑥∈Ω𝐹(𝑥,𝜉)/|𝜉|^𝑝0)}≤0, (𝐻₂)sup_𝑢∈𝑋∫_Ω𝐹(𝑥,𝑢(𝑥))𝑑𝑥>0. Then, for each compact interval [𝑎,𝑏]⊂(𝛾,∞), there exists 𝐵>0 with the following property: for every 𝜆∈[𝑎,𝑏] and 𝑔, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], the problem 𝑃_𝜆,𝜇 has at least three weak solutions whose norms in 𝑋 are less than 𝐵.

Proof of Theorem 3.1. In order to apply Lemma 2.5, we let 𝐼(𝑢)=_Ω[Φ(𝑥𝑢(𝑥𝐽(𝑢)=(|∇𝑢)|)+Φ(|)|)]𝑑𝑥,_Ω𝐹(𝑥,𝑢(𝑥))𝑑𝑥,Ψ(𝑢)=_Ω𝐺(𝑥,𝑢(𝑥))𝑑𝑥. (3.1) We divide our proof into two steps as follows.
Step 1. We show that some fundamental assumptions are satisfied.
𝑋∶=𝑊¹𝐿_Φ(Ω). Obviously, 𝑋 is a separable and reflexive real Banach space (see [13]). By Lemma 2.2, it is easy to see that 𝐼(𝑢) is a coercive, bounded on each bounded subset of 𝑋. On the other hand, 𝐼,𝐽,Ψ∈𝐶¹(𝑋,ℝ) with the derivatives given by 𝐼^′(𝑢),𝑣=_Ω[𝛼(𝑥(𝑥)(𝑥)𝑢(𝑥𝑢(𝑥)𝑣(𝑥𝐽(|∇𝑢)|)∇𝑢⋅∇𝑣+𝛼(|)|))]𝑑𝑥,_′(𝑢),𝑣=_Ω𝑓(𝑥,𝑢(𝑥))𝑣(𝑥)𝑑𝑥,Ψ^′(𝑢),𝑣=_Ω𝑔(𝑥,𝑢(𝑥))𝑣(𝑥)𝑑𝑥, (3.2) for any 𝑢,𝑣∈𝑋. Hence, the critical points of the functional 𝐼−𝜆𝐽−𝜇Ψ are exactly the weak solutions for problem 𝑃_𝜆,𝜇. Moreover, owing that Φ is convex, it follows that 𝐼 is convex. Hence, one has that 𝐼 is sequentially weakly lower semicontinuous. The fact 𝑋 is compactly embedded into 𝐶⁰(Ω) implies that operators 𝐽^′,Ψ^′ is compact. As the proof of Lemma 3.2 in [15], we know that 𝐼^′ has a continuous inverse.
Moreover, if {𝑢_𝑛} is a sequence in 𝑋 converging weakly to 𝑢 and liminf_𝑛→∞𝐼(𝑢_𝑛)≤𝐼(𝑢), remark that 𝐼 is sequentially weakly lower semicontinuous, one has 𝐼(𝑢)≤liminf_𝑛→∞𝐼𝑢_𝑛≤𝐼(𝑢). (3.3) Then, up to a subsequence, we deduce that 𝐼(𝑢_𝑛)→𝐼(𝑢)=𝑑. Taking into account that {(𝑢_𝑛+𝑢)/2} converges weakly to 𝑢 and 𝐼 is sequentially weakly lower semicontinuous, we have 𝑑=𝐼(𝑢)≤liminf_𝑛→∞𝐼𝑢_𝑛+𝑢2. (3.4) We assume by contradiction that 𝑢_𝑛 does not converge to 𝑢 in 𝑋. Hence, there exist 𝜀₀>0 and a subsequence {𝑢_𝑛𝑚} of (𝑢_𝑛) such that 𝑢_𝑛𝑚−𝑢2>𝜀₀,∀𝑚. (3.5) Then there exists 𝜀₁>0 such that 𝐼𝑢_𝑛𝑚−𝑢2>𝜀₁,∀𝑚. (3.6) On the other hand, 12𝐼(𝑢)+12𝐼𝑢_𝑛𝑚𝑢−𝐼^𝑛𝑚+𝑢2𝑢≥𝐼^𝑛𝑚−𝑢2>𝜀₁, (3.7)(see [19]). Letting 𝑚→∞ in the above inequality we obtain 𝑑−𝜀₁≥limsup_𝑚→∞𝐼𝑢_𝑛𝑚+𝑢2 (3.8) and that is a contradiction with (3.4). It follows that 𝑢_𝑛 converges strongly to 𝑢 and 𝐼∈𝒲_𝑋. In addition, 𝐼(0)=𝐽(0)=0.
Step 2. We show that 𝛼^′=0,𝛽^′>0.
In view of (𝐻₁), for all 𝜀>0, there exists 𝜏₁>0 such that 𝐹(𝑥,𝜉)≤𝜀𝜉^𝑝0, (3.9) for any |𝜉|∈[0,𝜏₁]. For ‖𝑢‖<min{1,(𝜏₁/2𝑐)}, we have |𝑢(𝑥)|≤‖𝑢‖_∞≤𝑐‖𝑢‖_1,Φ≤2𝑐‖𝑢‖≤𝜏₁,limsup_𝑢→0𝐽(𝑢)𝐼(𝑢)≤limsup_𝑢→0𝜀∫_Ω|𝑢|^𝑝0𝑑𝑥‖𝑢‖^𝑝0≤limsup_𝑢→0𝜀|Ω|(2𝑐‖𝑢‖)^𝑝0‖𝑢‖^𝑝0≤0. (3.10) By (𝐻₁), for all 𝜀>0, there exists 0<𝜏₁<𝜏₂ such that 𝐹(𝑥,𝜉)≤𝜀𝜉^𝑝0, (3.11) for any |𝜉|>𝜏₂. Further, for each ‖𝑢‖>1, we have 𝐽(𝑢)𝐼(𝑢)≤∫_{Ω(|𝑢|≤𝜏2)}𝐹(𝑥,𝑢(𝑥))𝑑𝑥‖𝑢‖^𝑝0+∫_{Ω(|𝑢|>𝜏2)}𝐹(𝑥,𝑢(𝑥))𝑑𝑥‖𝑢‖^𝑝0≤∫_{Ω(|𝑢|≤𝜏2)}𝐹(𝑥,𝑢(𝑥))𝑑𝑥‖𝑢‖^𝑝0+𝜀|Ω|(2𝑐‖𝑢‖)^𝑝0‖𝑢‖^𝑝0. (3.12) So we get limsup_{‖𝑢‖→∞}𝐽(𝑢)𝐼(𝑢)≤0. (3.13) Then, with the notation of Lemma 2.5, we have 𝛼^′=0. By assumption (𝐻₂), we have 𝛽^′>0. Thus, all the hypotheses of Lemma 2.5 are satisfied. Clearly, 𝛾=1/𝛽^′. Finally, by Lemma 2.5, we can obtain the Theorem 3.1.

Example 3.2. Let 𝑝>𝑁+1. Define 𝜑(𝑡)=|𝑡|^𝑝−2𝑡log(1+|𝑡|),𝑡≠0, (3.14) and 𝜑(0)=0. By [13], one has 𝑝₀=𝑝−1<𝑝⁰=𝑝. (3.15) Let 𝐹(𝑥,𝑡)=|𝑡|^𝑝+1−|𝑡|^𝑝+2. Since 𝐹≤0 if |𝑡| is large enough and 𝐹>0 if |𝑡| is small enough, moreover, it is easy to see limsup_𝜉→0(sup_𝑥∈Ω𝐹(𝑥,𝜉)/|𝜉|^𝑝0)=0, the conditions of Theorem 3.1 can be satisfied.

Remark 3.3. Since 𝐹 in [13] is 𝑝₀-sublinear, the results of [13] do not fit to the problem treated in the previous Example 3.2 even if 𝜇=0, that is, there is no perturbed nonlinear term. In addition, for nonhomogeneous Neumann problem with a perturbed term, we can have the following result when 𝐹 is 𝑝₀-sublinear, which extends the results of [13].

Theorem 3.4. Let Φ be a function satisfying the structural hypotheses (Φ₁–Φ₃) and the following conditions.
(𝐻₃) There exist two constants 𝛾,𝛿 with 𝛾<2𝑐 such that Φ(𝛿)>(𝛾^𝑝0/(2𝑐)^𝑝0|Ω|) and ∫_Ωmax_{|𝜉|≤𝛾}𝐹(𝑥,𝜉)𝑑𝑥𝛾^𝑝0<∫_Ω𝐹(𝑥,𝛿)𝑑𝑥(2𝑐)^𝑝0|Ω|Φ(𝛿), (3.16) where |Ω| denotes the Lebesgue measure of the set Ω.
(𝐻₄) There exist ℎ(𝑥),𝑘(𝑥)∈𝐿¹(Ω;ℝ⁺) and 0<𝑠<𝑝₀ such that |𝐹(𝑥,𝑡)|≤ℎ(𝑥)+𝑘(𝑥)|𝑡|^𝑠 for every (𝑥,𝑡)∈Ω×ℝ.
Then, there exist a nonempty open set 𝐴⊂[0,∞) and a positive number 𝐵, for each 𝜆∈𝐴 and for every 𝑔, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], the problem 𝑃_𝜆,𝜇 has at least three weak solutions whose norms in 𝑋 are less than 𝐵.

Proof of Theorem 3.4. Let us consider 𝐼,𝐽,Ψ as the proof of Theorem 3.1. For any 𝜆>0,𝑢∈𝑋, by (𝐻₄) we have 𝐼(𝑢)−𝜆𝐽(𝑢)≥_Ω[Φ(|𝑢(𝑥)|)+Φ(|∇𝑢(𝑥)|)]𝑑𝑥−𝜆‖𝑢‖^𝑠_∞Ω𝑘(𝑥)𝑑𝑥−𝜆_Ωℎ(𝑥)≥‖𝑢‖𝑑𝑥_𝑝0−𝜆(2𝑐)^𝑠‖𝑢‖^𝑠_Ω𝑘(𝑥)𝑑𝑥−𝜆_Ωℎ(𝑥)𝑑𝑥, (3.17) for ‖𝑢‖>1. Since 0<𝑠<𝑝₀, one has lim_{‖𝑢‖→∞}[𝐼(𝑢)−𝜆𝐽(𝑢)]=+∞ for all 𝜆≥0.
Let 𝑟∶=(𝛾^𝑝0/(2𝑐)^𝑝0),   𝑥₁=𝛿. For 𝑟>0,𝐼(𝑢)≤𝑟, we have |𝑢(𝑥)|≤‖𝑢‖_∞≤𝑐‖𝑢‖_1,Φ≤2𝑐‖𝑢‖≤𝛾,∀𝑥∈Ω. (3.18) Hence, one has sup_{𝑢∈𝐼−1(]−∞,𝑟])}𝐽(𝑢)𝑟≤(2𝑐)^𝑝0∫_Ωmax_{|𝜉|≤𝛾}𝐹(𝑥,𝜉)𝑑𝑥𝛾^𝑝0. (3.19) From (𝐻₃), it follows that sup_{𝐼(𝑢)≤𝑟}𝐽(𝑢)𝑟<𝐽(𝛿)𝐼(𝛿). (3.20) Since all the assumptions of Lemmas 2.7 and 2.6 are satisfied, then, there is a nonempty open set 𝐴⊂[0,∞) and a positive number 𝐵, for each 𝜆∈𝐴 and for every 𝑔 there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], the problem has at least three weak solutions whose norms in 𝑋 are less than 𝐵.

Acknowledgments

L. Yang was supported by the Natural Science Foundation of Hunan Province (12JJ9001), Hunan Provincial Science and Technology Department of Science and Technology Project (2012SK3117), and Hengyang Normal University Doctoral Science Foundation (11B35).