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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 749059, 14 pages
http://dx.doi.org/10.1155/2012/749059
Research Article

Multiplicity of Solutions for a Class of Fourth-Order Elliptic Problems with Asymptotically Linear Term

School of Mathematics and Statistics, Hubei Engineering University, Hubei, Xiaogan 432000, China

Received 10 January 2012; Accepted 5 April 2012

Academic Editor: TurgutΒ Γ–ziş

Copyright Β© 2012 Qiong Liu and Dengfeng LΓΌ. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the following fourth-order elliptic equations: Ξ”2𝑒+π‘ŽΞ”π‘’=𝑓(π‘₯,𝑒),π‘₯∈Ω,𝑒=Δ𝑒=0,π‘₯βˆˆπœ•Ξ©, where Ξ©βŠ‚β„π‘ is a bounded domain with smooth boundary πœ•Ξ© and 𝑓(π‘₯,𝑒) is asymptotically linear with respect to 𝑒 at infinity. Using an equivalent version of Cerami's condition and the symmetric mountain pass lemma, we obtain the existence of multiple solutions for the equations.

1. Introduction and Main Results

In this paper, we will investigate the existence of multiple solutions to the following fourth-order elliptic boundary value problem: Ξ”2𝑒+π‘ŽΞ”π‘’=𝑓(π‘₯,𝑒),π‘₯∈Ω,𝑒=Δ𝑒=0,π‘₯βˆˆπœ•Ξ©,(1.1) where Ξ©βŠ‚β„π‘ is a bounded domain with smooth boundary πœ•Ξ©, Ξ”2 is the biharmonic operator, π‘Ž<πœ†1 (πœ†1 is the first eigenvalue of βˆ’Ξ” in 𝐻10(Ξ©)) is a parameter. We assume that 𝑓(π‘₯,𝑒) satisfies the following hypotheses.(𝑓1)𝑓(π‘₯,𝑒)∈𝐢(Ω×ℝ,ℝ).(𝑓2)lim|𝑒|β†’0𝑓(π‘₯,𝑒)/𝑒=0 uniformly for π‘₯∈Ω.(𝑓3)lim|𝑒|β†’βˆžπ‘“(π‘₯,𝑒)/𝑒=β„“ uniformly for π‘₯∈Ω, where β„“βˆˆ(0,+∞) is a constant, or β„“=+∞, and there exists 𝐢>0, π‘žβˆˆ[2,2βˆ—) such that ||||𝑓(π‘₯,𝑒)≀𝐢1+|𝑒|π‘žβˆ’1ξ€Έ,(1.2)where 2βˆ—=2𝑁/(π‘βˆ’4).(𝑓4)𝑓(π‘₯,𝑒) is odd in 𝑒.(𝑓5)lim|𝑒|β†’βˆž(𝑓(π‘₯,𝑒)π‘’βˆ’2𝐹(π‘₯,𝑒))=+∞ uniformly for π‘₯∈Ω, where∫𝐹(π‘₯,𝑒)=𝑒0𝑓(π‘₯,𝑑)𝑑𝑑.(𝑓6)𝑓(π‘₯,𝑒)/𝑒 is nondecreasing with respect to 𝑒β‰₯0, for a.e. π‘₯∈Ω.

Problem (1.1) is usually used to describe some phenomena appeared in different physical, engineering and other sciences. In recent years, there are many results for the fourth-order elliptic equations. In [1], Lazer and McKenna considered the fourth-order problem: Ξ”2ξ€·(𝑒+π‘ŽΞ”π‘’=𝑑𝑒+1)+ξ€Έβˆ’1,π‘₯∈Ω,𝑒=Δ𝑒=0,π‘₯βˆˆπœ•Ξ©,(1.3) where 𝑒+=max{𝑒,0} and π‘‘βˆˆβ„. They pointed out that this type of nonlinearity furnishes a model to study traveling waves in suspension bridges. They also presented a mathematical model for the bridge that takes account of the fact that the coupling provided by the stays connecting the suspension cable to the deck of the road bed is fundamentally nonlinear (see [1–3]). Since then, more general nonlinear fourth-order elliptic boundary value problems have been studied. Problem (1.1) and (1.3) have been studied extensively in recent years, we refer the reader to [4–14].

For problem (1.3), Lazer and McKenna [2] proved the existence of 2π‘˜βˆ’1 solutions when 𝑁=1 and 𝑑>πœ†π‘˜(πœ†π‘˜βˆ’π‘) (πœ†π‘˜ is the sequence of the eigenvalues of βˆ’Ξ” in 𝐻10(Ξ©)) by the global bifurcation method. In [4], Tarantello found a negative solution when 𝑑>πœ†1(πœ†1βˆ’π‘) by a degree argument. For Problem (1.1), when 𝑓(π‘₯,𝑒)=𝑏𝑔(π‘₯,𝑒), the existence of two or three nontrivial solutions has been obtained in [5, 6] for 𝑔(π‘₯,𝑒) under certain conditions by using variational methods. In [7], positive solutions of problem (1.1) were got when 𝑓 satisfies the local superlinearity and sublinearity. When 𝑓 is asymptotically linear at infinity, the existence of three nontrivial solutions has been obtained in [8] by using variational method, and the existence of a nontrivial solution has been obtained in [9] by using the mountain pass theorem. For more similar problems, we refer to [10–20] and the references therein.

In this paper, we prove a new existence result about a multiple solutions of problem (1.1) under the assumption that 𝑓(π‘₯,𝑒) is asymptotically linear with respect to 𝑒 at infinity. In this case, the Ambrosetti-Rabinowitz condition ((AR) condition for short) does not hold, hence it is difficult to verify the classical (PS)𝑐 condition. To overcome this difficulty, by using an equivalent version of Cerami’s condition and the symmetric mountain pass lemma (see [21]), we obtain the existence of multiple solutions for problem (1.1). To the best of our knowledge, our main results are new. Before stating the main results, we give some notations.

Set 𝐸=𝐻2(Ξ©)∩𝐻10(Ξ©), then 𝐸 is a Hilbert space with the following inner product and the norm: βŸ¨π‘’,π‘£βŸ©πΈ=ξ€œΞ©(Ξ”π‘’Ξ”π‘£βˆ’π‘Žβˆ‡π‘’βˆ‡π‘£)𝑑π‘₯,‖𝑒‖𝐸=βŸ¨π‘’,π‘’βŸ©πΈ1/2.(1.4)

The corresponding energy functional of problem (1.1) is defined on 𝐸 by 1𝐼(𝑒)=2ξ€œΞ©ξ‚€||||Δ𝑒2||||βˆ’π‘Žβˆ‡π‘’2ξ‚ξ€œπ‘‘π‘₯βˆ’Ξ©πΉ(π‘₯,𝑒)𝑑π‘₯,(1.5) where ∫𝐹(π‘₯,𝑒)=𝑒0𝑓(π‘₯,𝑑)𝑑𝑑. From (𝑓1)–(𝑓3), it is easy to see that 𝐼∈𝐢1(𝐸,ℝ), it is well known that the weak solutions of problem (1.1) are the critical points of the energy functional 𝐼(𝑒).

Our main results are stated as follows.

Theorem 1.1. Assume that 𝑓(π‘₯,𝑒) satisfies assumptions (𝑓1)–(𝑓4), and Ξ›π‘˜ is given by (2.8). Then the following hold. (i)If β„“βˆˆ(Ξ›π‘˜,+∞) is not an eigenvalue of problem (2.4), then problem (1.1) has at least π‘˜ pairs of nontrivial solutions in 𝐸.(ii)Suppose that (𝑓5) is satisfied, then the conclusion of (i) holds even if β„“ is an eigenvalue of problem (2.4).(iii)If β„“=+∞, and (𝑓6) holds, then problem (1.1) has infinitely many nontrivial solutions in 𝐸.

2. Preliminaries

In this section, we give some preliminary results which will be used to prove our main results.

Throughout this paper, we will denote by |Ξ©| the Lebesgue measure of Ξ©, 𝐡𝜌={π‘’βˆˆπΈβˆΆβ€–π‘’β€–πΈ<𝜌}. 𝐢 will denote various positive constants, β†’ (respectively⇀) denotes strong (respectively weak) convergence. π‘œπ‘š(1) denote π‘œπ‘š(1)β†’0 as π‘šβ†’βˆž. 𝐿𝑠(Ξ©),(1≀𝑠<+∞) denote Lebesgue spaces, the norm 𝐿𝑠 is denoted by |β‹…|𝑠 for 1≀𝑠<+∞. The dual space of a Banach space 𝐸 will be denoted by πΈβˆ’1.

First, we recall an equivalent version of Cerami’s condition as follows (see [22]).

Definition 2.1. Let 𝐸 be a Banach space. 𝐼∈𝐢1(𝐸,ℝ) is said to satisfy condition (𝐢) at level π‘βˆˆβ„ ((𝐢)𝑐 for short), if the following fact is true: any sequence {π‘’π‘š}βŠ‚πΈ, which satisfiesπΌξ€·π‘’π‘šξ€Έξ€·βŸΆπ‘,1+β€–π‘’π‘šβ€–πΈξ€Έβ€–πΌξ…žξ€·π‘’π‘šξ€Έβ€–πΈβˆ’1⟢0,(π‘šβŸΆβˆž)(2.1) possesses a convergent subsequence in 𝐸.

Next, we will state an abstract symmetric mountain pass lemma. For this purpose, we should first introduce the definition of genus (see [23–25]).

Definition 2.2. Let 𝐸 be a real Banach space and 𝐴 a subset of 𝐸. 𝐴 is said to be symmetric if π‘’βˆˆπ΄ implies βˆ’π‘’βˆˆπ΄. For a closed symmetric set 𝐴 which does not contain the origin, we define a genus 𝛾(𝐴) of 𝐴 by the smallest integer π‘˜ such that there exists an odd continuous mapping from 𝐴 to β„π‘˜β§΅{0}. If there does not exist such a π‘˜, we define 𝛾(𝐴)=∞. Moreover, we set 𝛾(βˆ…)=0.

Let 𝐸 be an infinite dimensional real Banach space, β€‰πΌβˆˆπΆ1(𝐸,ℝ), 𝐴0={π‘’βˆˆπΈβˆΆπΌ(𝑒)β‰₯0}, Ξ“βˆ—={β„Ž(0)=0,β„Žisanoddhomeomorphismof𝐸andβ„Ž(𝐡1𝐴)βŠ‚0}, Ξ“π‘š={π’¦βŠ‚πΈβˆΆπ’¦ is compact, symmetric with respect to the origin, and for any β„ŽβˆˆΞ“βˆ—, there holds 𝛾(π’¦βˆ©β„Ž(πœ•π΅1))β‰₯π‘š}. If Ξ“π‘šβ‰ βˆ…, defineπ‘π‘š=infπ’¦βŠ‚Ξ“π‘šmaxπ‘’βˆˆπ’¦πΌ(𝑒).(2.2)

Now, we recall an abstract symmetric mountain pass lemma, which can be found in [26, 27].

Lemma 2.3. Let 𝑒1,𝑒2,…,π‘’π‘š,… be linearly independent in 𝐸, and 𝐸𝑖=span{𝑒1,𝑒2,…,𝑒𝑖}, 𝑖=1,2,…,π‘š,…. Suppose that 𝐼∈𝐢1(𝐸,ℝ) satisfies 𝐼(0)=0, 𝐼(βˆ’π‘’)=𝐼(𝑒), and (𝐢)𝑐 condition for 𝑐β‰₯0. Furthermore, there exists 𝜌>0, 𝛼>0 such that 𝐼(𝑒)>0 in 𝐡𝜌⧡{0} and 𝐼(𝑒)|πœ•π΅πœŒβ‰₯𝛼. Then, if πΈπ‘šβˆ©ξπ΄0 is bounded, then Ξ“π‘šβ‰ βˆ… and π‘π‘šβ‰₯𝛼>0 is a critical value of 𝐼. Moreover, if πΈπ‘š+π‘–βˆ©ξπ΄0 is bounded for all 𝑖=1,β€¦π‘Ÿ, andπ‘π‘š+1=β‹―=π‘π‘š+π‘Ÿ=𝑏,(2.3) then 𝛾(𝐾𝑏)β‰₯π‘Ÿ, where 𝐾𝑏={π‘’βˆˆπΈβˆΆπΌ(𝑒)=𝑏,πΌξ…ž(𝑒)=0}. If πΈπ‘šβˆ©ξπ΄0 is bounded for all π‘š, then 𝐼(𝑒) possesses infinitely many critical values.

Let us consider the eigenvalue problem:Ξ”2𝑒+π‘ŽΞ”π‘’=Λ𝑒,π‘₯∈Ω,𝑒=Δ𝑒=0,π‘₯βˆˆπœ•Ξ©.(2.4)Set ξ€œΞ¦(𝑒)=Ξ©ξ‚€||||Δ𝑒2||||βˆ’π‘Žβˆ‡π‘’2ξ‚ξ€œπ‘‘π‘₯,Ξ¨(𝑒)=Ξ©|𝑒|2𝑑π‘₯.(2.5)For π‘Ž<πœ†1, Ξ¦(𝑒) and Ξ¨(𝑒) are well defined. Furthermore, Ξ¦(𝑒),Ξ¨(𝑒)∈𝐢1(𝐸,ℝ), and a real value Ξ› is an eigenvalue of problem (2.4) if and only if there exists π‘’βˆˆπΈβ§΅{0} such that Ξ¦ξ…ž(𝑒)=Ξ›Ξ¨ξ…ž(𝑒). At this point, let us set 𝒩={π‘’βˆˆπΈβˆΆΞ¨(𝑒)=1}.(2.6)Then π’©β‰ βˆ… and 𝒩 is a 𝐢1 manifold in 𝐸. It follows from the standard Lagrange multiples arguments that eigenvalues of (2.4) correspond to critical values of Ξ¦|𝒩, and Ξ¦ satisfies the (PS) condition on 𝒩. Thus a sequence of critical values of Ξ¦|𝒩 comes from the Ljusternik-Schnirelmann critical point theory on 𝐢1 manifolds. For any π‘˜βˆˆπ‘, set Ξ“π‘˜={π΄βŠ‚π’©βˆΆπ΄iscompact,symmetricand𝛾(𝐴)β‰₯π‘˜}.(2.7) Then values: Ξ›π‘˜βˆΆ=infπ΄βˆˆΞ“π‘˜maxπ‘’βˆˆπ΄Ξ¦(𝑒)(2.8)are critical values and hence are eigenvalues of problem (2.4). Moreover, 0<Ξ›1<Ξ›2≀Λ3β‰€β‹―β‰€Ξ›π‘˜β‰€β‹―β†’+∞.

We prove some properties of functional 𝐼(𝑒) in the following lemma.

Lemma 2.4. For the functional 𝐼(𝑒) defined by (1.5), if assumptions (𝑓1) and (𝑓6) hold, and for any {π‘’π‘š}βŠ‚πΈ with βŸ¨πΌξ…ž(π‘’π‘š),π‘’π‘šβŸ©β†’0 as π‘šβ†’βˆž, then there is a subsequence, still denoted by {π‘’π‘š}, such thatπΌξ€·π‘‘π‘’π‘šξ€Έβ‰€1+𝑑2𝑒2π‘š+πΌπ‘šξ€Έ(2.9) holds for all 𝑑>0, π‘šβˆˆπ‘+.

Proof. This lemma is essentially due to [27, 28]. For the sake of completeness, we prove it here.
By βŸ¨πΌβ€²(π‘’π‘š),π‘’π‘šβŸ©β†’0 as π‘šβ†’βˆž, for a suitable subsequence, we may assume that βˆ’1π‘š<ξ«ξ€·π‘’πΌβ€²π‘šξ€Έ,π‘’π‘šξ¬=β€–β€–π‘’π‘šβ€–β€–2πΈβˆ’ξ€œΞ©π‘“ξ€·π‘₯,π‘’π‘šξ€Έπ‘’π‘š1𝑑π‘₯<π‘š,βˆ€π‘š.(2.10) We claim that for any 𝑑>0 and π‘šβˆˆπ‘+, πΌξ€·π‘‘π‘’π‘šξ€Έ<𝑑2+ξ€œ2π‘šΞ©ξ‚€12𝑓π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ€·βˆ’πΉπ‘₯,π‘’π‘šξ€Έξ‚π‘‘π‘₯.(2.11) Indeed, for any 𝑑>0, at fixed π‘₯∈Ω and π‘šβˆˆπ‘+, we set π‘‘β„Ž(𝑑)=22𝑓π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ€·βˆ’πΉπ‘₯,π‘‘π‘’π‘šξ€Έ,(2.12) then β„Žξ…žξ€·(𝑑)=𝑑𝑓π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ€·βˆ’π‘“π‘₯,π‘‘π‘’π‘šξ€Έπ‘’π‘š=π‘‘π‘’π‘šξ‚€π‘“ξ€·π‘₯,π‘’π‘šξ€Έβˆ’1𝑑𝑓π‘₯,π‘‘π‘’π‘šξ€Έξ€·π‘“ξ‚ξ‚†β‰₯0,0<𝑑≀1,≀0,𝑑β‰₯1,by6ξ€Έ,(2.13) hence β„Ž(𝑑)β‰€β„Ž(1)βˆ€π‘‘>0.(2.14) Therefore, πΌξ€·π‘‘π‘’π‘šξ€Έ=𝑑22β€–β€–π‘’π‘šβ€–β€–2πΈβˆ’ξ€œΞ©πΉξ€·π‘₯,π‘‘π‘’π‘šξ€Έβ‰€π‘‘π‘‘π‘₯22ξ‚΅1π‘š+ξ€œΞ©π‘“ξ€·π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ‚Άβˆ’ξ€œπ‘‘π‘₯Ω𝐹π‘₯,π‘‘π‘’π‘šξ€Έβ‰€π‘‘π‘‘π‘₯by(2.10)2+ξ€œ2π‘šΞ©ξ‚΅π‘‘22𝑓π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ€·βˆ’πΉπ‘₯,π‘‘π‘’π‘šξ€Έξ‚Άβ‰€π‘‘π‘‘π‘₯2+ξ€œ2π‘šΞ©ξ‚€12𝑓π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ€·βˆ’πΉπ‘₯,π‘’π‘šξ€Έξ‚π‘‘π‘₯by(2.14)(2.15)and our claim (2.11) is proved.
On the other hand, πΌξ€·π‘’π‘šξ€Έ=12β€–β€–π‘’π‘šβ€–β€–2πΈβˆ’ξ€œΞ©πΉξ€·π‘₯,π‘’π‘šξ€Έβ‰₯1𝑑π‘₯2ξ‚΅βˆ’1π‘š+ξ€œΞ©π‘“ξ€·π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ‚Άβˆ’ξ€œπ‘‘π‘₯Ω𝐹π‘₯,π‘’π‘šξ€Έπ‘‘π‘₯,(2.16) that is, ξ€œΞ©ξ‚€12𝑓π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ€·βˆ’πΉπ‘₯,π‘’π‘šξ€Έξ‚1𝑑π‘₯≀𝑒2π‘š+πΌπ‘šξ€Έ.(2.17) Combining (2.11) and (2.17) we have that πΌξ€·π‘‘π‘’π‘šξ€Έβ‰€1+𝑑2𝑒2π‘š+πΌπ‘šξ€Έ,βˆ€π‘‘>0,π‘šβˆˆπ‘+.(2.18)
The proof is completed.

3. Proof of the Main Results

We begin with the following lemma.

Lemma 3.1. Let 𝑐β‰₯0. Assume that 𝑓(π‘₯,𝑒) satisfies assumptions (𝑓1)–(𝑓3). Then the following hold. (i)𝐼(𝑒) satisfies (𝐢)𝑐 condition if β„“<+∞ in assumption (𝑓3), and β„“ is not an eigenvalue of problem (2.4).(ii)If β„“<+∞ is an eigenvalue of problem (2.4) and (𝑓5) holds, then 𝐼(𝑒) satisfies (𝐢)𝑐 condition.(iii)If β„“=+∞, and (𝑓6) holds, then 𝐼(𝑒) satisfies (𝐢)𝑐 condition.

Proof. Suppose that {π‘’π‘š}βŠ‚πΈ is a (𝐢)𝑐 sequence, that is, as π‘šβ†’βˆž, we have πΌξ€·π‘’π‘šξ€Έξ€·β€–β€–π‘’βŸΆπ‘β‰₯0,(3.1)1+π‘šβ€–β€–πΈξ€Έβ€–β€–πΌβ€²(π‘’π‘š)β€–β€–πΈβˆ’1⟢0,inπΈβˆ’1.(3.2) It is easy to see that (3.2) implies that as π‘šβ†’βˆž, there hold β€–β€–π‘’π‘šβ€–β€–2πΈβˆ’ξ€œΞ©π‘“ξ€·π‘₯,π‘’π‘šξ€Έπ‘’π‘šπ‘‘π‘₯=π‘œπ‘š(ξ€œ1),(3.3)Ξ©ξ€·Ξ”π‘’π‘šΞ”πœ‘βˆ’π‘Žβˆ‡π‘’π‘šξ€Έξ€œβˆ‡πœ‘π‘‘π‘₯βˆ’Ξ©π‘“ξ€·π‘₯,π‘’π‘šξ€Έπœ‘π‘‘π‘₯=π‘œπ‘š(1),βˆ€πœ‘βˆˆπΈ.(3.4) By Sobolev compact embedding, to show that 𝐼(𝑒) satisfies (𝐢)𝑐 condition, it suffices to show the boundedness of (𝐢)𝑐 sequence in 𝐸 for each case.
(i) Suppose that 0<β„“<+∞ and β„“ is not an eigenvalue of problem (2.4). Arguing by contradiction, we suppose that there exists a subsequence, still denoted by {π‘’π‘š}, such that as π‘šβ†’βˆž, there holds β€–π‘’π‘šβ€–πΈβ†’+∞. Define π‘π‘šβŽ§βŽͺ⎨βŽͺβŽ©π‘“ξ€·(π‘₯)=π‘₯,π‘’π‘šξ€Έ(π‘₯)π‘’π‘š(π‘₯),π‘’π‘š(π‘₯)β‰ 0,0,π‘’π‘š(π‘₯)=0.(3.5) Then from assumptions (𝑓1)–(𝑓3), there exists 𝑀>0 such that 0β‰€π‘π‘š(π‘₯)≀𝑀.(3.6) Let π‘€π‘š=π‘’π‘šβ€–π‘’π‘šβ€–πΈ.(3.7) Obviously, π‘€π‘š is bounded in 𝐸. Going if necessary to a subsequence, we can assume that π‘€π‘šπ‘€β‡€π‘€,weaklyin𝐸,π‘šπ‘€βŸΆπ‘€,a.e.inΞ©,π‘šβŸΆπ‘€,stronglyin𝐿𝑠(Ξ©),βˆ€π‘ βˆˆ2,2βˆ—ξ€Έ.(3.8) It is easy to show that 𝑀≒0. In fact, if 𝑀≑0, then from (3.3), (3.6), (3.8) and the definitions of π‘π‘š and π‘€π‘š, as π‘šβ†’βˆž, we have ‖‖𝑀1=π‘šβ€–β€–2𝐸=ξ€œΞ©π‘π‘š(||𝑀π‘₯)π‘š||2𝑑π‘₯+π‘œπ‘š(ξ€œ1)≀𝑀Ω||π‘€π‘š||2𝑑π‘₯+π‘œπ‘š(1)⟢0,(3.9) which is a contradiction.
From (3.6), there exists β„Ž(π‘₯)∈𝐿∞(Ξ©) with 0β‰€β„Ž(π‘₯)≀𝑀 such that, up to a subsequence, as π‘šβ†’βˆž, there holds π‘π‘š(π‘₯)β‡€β„Ž(π‘₯),weaklyβˆ—in𝐿∞(Ξ©).(3.10) Then from (3.8) it follows that π‘π‘š(π‘₯)π‘€π‘šβ‡€β„Ž(π‘₯)𝑀weaklyin𝐿2ξ€œ(Ξ©),Ξ©π‘π‘š||𝑀(π‘₯)π‘š||2ξ€œπ‘‘π‘₯βŸΆΞ©β„Ž(π‘₯)|𝑀|2𝑑π‘₯.(3.11) On the other hand, from (3.3), (3.4), (3.5), and (3.7), we have ξ€œΞ©ξ€·Ξ”π‘€π‘šΞ”πœ‘βˆ’π‘Žβˆ‡π‘€π‘šξ€Έξ€œβˆ‡πœ‘π‘‘π‘₯=Ξ©π‘π‘š(π‘₯)π‘€π‘šπœ‘π‘‘π‘₯+π‘œπ‘š(1),βˆ€πœ‘βˆˆπΈ.(3.12)ξ€œΞ©ξ‚€||Ξ”π‘€π‘š||2||βˆ’π‘Žβˆ‡π‘€π‘š||2ξ‚ξ€œπ‘‘π‘₯=Ξ©π‘π‘š(||𝑀π‘₯)π‘š||2𝑑π‘₯+π‘œπ‘š(1).(3.13) It follows from (3.11)–(3.13) that β€–β€–π‘€π‘šβ€–β€–2𝐸=ξ€œΞ©β„Ž(π‘₯)|𝑀|2𝑑π‘₯+π‘œπ‘š(ξ€œ1),Ξ©ξ€·Ξ”π‘€π‘šΞ”πœ‘βˆ’π‘Žβˆ‡π‘€π‘šξ€Έξ€œβˆ‡πœ‘π‘‘π‘₯=Ξ©β„Ž(π‘₯)π‘€πœ‘π‘‘π‘₯+π‘œπ‘š(1),βˆ€πœ‘βˆˆπΈ.(3.14) Therefore (3.14) implies that 𝑀 satisfies ξ€œΞ©(ξ€œΞ”π‘€Ξ”πœ‘βˆ’π‘Žβˆ‡π‘€βˆ‡πœ‘)𝑑π‘₯=Ξ©β„Ž(π‘₯)π‘€πœ‘π‘‘π‘₯,βˆ€πœ‘βˆˆπΈ.(3.15) Let Ξ©0=Ξ©{π‘₯βˆˆΞ©βˆΆπ‘€(π‘₯)=0},+Ξ©={π‘₯βˆˆΞ©βˆΆπ‘€(π‘₯)>0},βˆ’={π‘₯βˆˆΞ©βˆΆπ‘€(π‘₯)<0}.(3.16) Then π‘’π‘š(π‘₯)β†’+∞ as π‘šβ†’βˆž if π‘₯∈Ω+, and π‘’π‘š(π‘₯)β†’βˆ’βˆž as π‘šβ†’βˆž if π‘₯βˆˆΞ©βˆ’. From assumption (𝑓3), β„Ž(π‘₯)≑ℓ for all π‘₯∈Ω+βˆͺΞ©βˆ’. Thus (3.15) implies that 𝑀 satisfies ξ€œΞ©0(ξ€œΞ”π‘€Ξ”πœ‘βˆ’π‘Žβˆ‡π‘€βˆ‡πœ‘βˆ’β„Ž(π‘₯)π‘€πœ‘)𝑑π‘₯+Ξ©+βˆͺΞ©βˆ’(Ξ”π‘€Ξ”πœ‘βˆ’π‘Žβˆ‡π‘€βˆ‡πœ‘βˆ’β„Ž(π‘₯)π‘€πœ‘)𝑑π‘₯=0,βˆ€πœ‘βˆˆπΈ.(3.17) Therefore ξ€œΞ©(ξ€œΞ”π‘€Ξ”πœ‘βˆ’π‘Žβˆ‡π‘€βˆ‡πœ‘)𝑑π‘₯=β„“Ξ©π‘€πœ‘π‘‘π‘₯,βˆ€πœ‘βˆˆπΈ.(3.18) This means that β„“ is an eigenvalue of problem (2.4), which contradicts our assumption, so {π‘’π‘š} is bounded in 𝐸.
(ii) Suppose β„“βˆˆ(0,+∞) is an eigenvalue of problem (2.4), we need the additional assumption (𝑓5).
From assumption (𝑓5), there exists 𝑇0>0 such that𝑓(π‘₯,𝑒)π‘’βˆ’2𝐹(π‘₯,𝑒)β‰₯0,βˆ€|𝑒|β‰₯𝑇0,π‘₯∈Ω,(3.19) and there exists 𝐢0=𝐢0(𝑇0)>0 such that ξ€œ{|π‘’π‘š|≀𝑇0}𝑓π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ€·βˆ’2𝐹π‘₯,π‘’π‘šξ€Έξ€Έπ‘‘π‘₯β‰₯βˆ’πΆ0.(3.20) Furthermore, under assumptions (𝑓1)–(𝑓3), there exists 𝑀>0 such that ||||||||≀𝑀𝑓(π‘₯,𝑒)≀𝑀|𝑒|,𝐹(π‘₯,𝑒)2|𝑒|2,βˆ€π‘₯∈Ω.(3.21) Let 𝐾=(2𝑐+𝐢0)(2𝑀𝑆)𝑁/2, where 𝑀>0 is given by (3.21), 𝑆>0 is the best Sobolev constant such that ξ‚΅ξ€œΞ©|𝑒|2βˆ—ξ‚Άπ‘‘π‘₯2/2βˆ—ξ€œβ‰€π‘†Ξ©ξ‚€||||Δ𝑒2||||βˆ’π‘Žβˆ‡π‘’2𝑑π‘₯,βˆ€π‘’βˆˆπΈ.(3.22) From assumption (𝑓5), there exists 𝑇=𝑇(𝐾)>𝑇0>0 such that 𝑓(π‘₯,𝑒)π‘’βˆ’2𝐹(π‘₯,𝑒)β‰₯𝐾,βˆ€|𝑒|β‰₯𝑇,π‘₯∈Ω.(3.23) For the above 𝑇>0 and each π‘šβ‰₯1, set π΄π‘š=ξ€½||𝑒π‘₯βˆˆΞ©βˆΆπ‘š||ξ€Ύ(π‘₯)β‰₯𝑇,π΅π‘š=ξ€½||𝑒π‘₯βˆˆΞ©βˆΆπ‘š||ξ€Ύ(π‘₯)≀𝑇.(3.24) From estimates (3.20), (3.1), (3.3), and (3.23), we get 2𝑐+π‘œπ‘š(ξ€œ1)=Ω𝑓π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ€·βˆ’2𝐹π‘₯,π‘’π‘šβ‰₯ξ€œξ€Έξ€Έπ‘‘π‘₯π΄π‘šξ€·π‘“ξ€·π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ€·βˆ’2𝐹π‘₯,π‘’π‘šξ€Έξ€Έπ‘‘π‘₯βˆ’πΆ0||𝐴β‰₯πΎπ‘š||βˆ’πΆ0,(3.25) where |π΄π‘š| denotes the measure of π΄π‘š.
On the other hand, for any fixed π‘Ÿ>2, from (3.1) and (3.3), we haveξ‚€12βˆ’1π‘Ÿξ‚β€–β€–π‘’π‘šβ€–β€–2πΈβˆ’ξ€œΞ©ξ‚€πΉξ€·π‘₯,π‘’π‘šξ€Έβˆ’1π‘Ÿπ‘“ξ€·π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ‚π‘‘π‘₯=𝑐+π‘œπ‘š(1).(3.26) Since Ξ© is bounded and π‘“βˆˆπΆ(Ω×ℝ,ℝ), there exists a constant 𝐢=𝐢(Ξ©,𝑓,𝑇) such that ||||ξ€œπ΅π‘šξ‚€πΉξ€·π‘₯,π‘’π‘šξ€Έβˆ’1π‘Ÿπ‘“ξ€·π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ‚||||𝑑π‘₯≀𝐢,βˆ€π‘₯∈Ω.(3.27) Then, from (3.21)–(3.26), HΓΆlder inequality and Sobolev inequality, we have 𝑐+π‘œπ‘š(ξ‚€11)β‰₯2βˆ’1π‘Ÿξ‚β€–β€–π‘’π‘šβ€–β€–2πΈξ€œβˆ’πΆβˆ’π΄π‘šξ‚€πΉξ€·π‘₯,π‘’π‘šξ€Έβˆ’1π‘Ÿπ‘“ξ€·π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ‚β‰₯ξ‚€1𝑑π‘₯2βˆ’1π‘Ÿξ‚β€–β€–π‘’π‘šβ€–β€–2πΈξ€œβˆ’πΆβˆ’π΄π‘šξ‚€12𝑓π‘₯,π‘’π‘šξ€Έπ‘’π‘šβˆ’1π‘Ÿπ‘“ξ€·π‘₯,π‘’π‘šξ€Έπ‘’π‘šξ‚=ξ‚€1𝑑π‘₯2βˆ’1π‘Ÿξ‚β€–β€–π‘’π‘šβ€–β€–2𝐸1βˆ’πΆβˆ’2βˆ’1π‘Ÿξ‚ξ€œπ΄π‘šπ‘“ξ€·π‘₯,π‘’π‘šξ€Έπ‘’π‘šβ‰₯ξ‚€1𝑑π‘₯2βˆ’1π‘Ÿξ‚β€–β€–π‘’π‘šβ€–β€–2𝐸1βˆ’πΆβˆ’2βˆ’1π‘Ÿξ‚π‘€ξ€œπ΄π‘š||π‘’π‘š||2β‰₯ξ‚€1𝑑π‘₯2βˆ’1π‘Ÿξ‚β€–β€–π‘’π‘šβ€–β€–2𝐸1βˆ’πΆβˆ’2βˆ’1π‘Ÿξ‚π‘€||π‘’π‘š||22βˆ—||π΄π‘š||2/𝑁β‰₯ξ‚€12βˆ’1π‘Ÿξ‚β€–β€–π‘’π‘šβ€–β€–2𝐸1βˆ’πΆβˆ’2βˆ’1π‘Ÿξ‚β€–β€–π‘’π‘€π‘†π‘šβ€–β€–2𝐸||||2𝑐+𝐢0𝐾||||+π‘œπ‘šξƒͺ(1)2/𝑁β‰₯12ξ‚€12βˆ’1π‘Ÿξ‚β€–β€–π‘’π‘šβ€–β€–2𝐸1βˆ’πΆβˆ’2βˆ’1π‘Ÿξ‚β€–β€–π‘’π‘€π‘†π‘šβ€–β€–2πΈβ‹…π‘œπ‘š(1),(3.28) that is, {π‘’π‘š} is bounded in 𝐸.
(iii) Finally, we prove the case β„“=+∞. Here the subcritical condition (1.2) is assumer as usual, but to make use of Lemma 2.4, (𝑓6) is required in this case. Set π‘‘π‘š=2βˆšπ‘β€–β€–π‘’π‘šβ€–β€–πΈ,π‘€π‘š=π‘‘π‘šπ‘’π‘š=2βˆšπ‘π‘’π‘šβ€–β€–π‘’π‘šβ€–β€–πΈ.(3.29) Then β€–π‘€π‘šβ€–πΈβˆš=2𝑐 and {π‘€π‘š} is bounded in 𝐸. Hence, up to a subsequence, we may assume that: there exists π‘€βˆˆπΈ such that (3.8) also holds in this case. If β€–π‘’π‘šβ€–πΈβ†’+∞, we claim that 𝑀(π‘₯)β‰’0.(3.30) In fact, if 𝑀(π‘₯)≑0 in Ξ©, then (3.29) and (3.8) imply that ξ€œΞ©πΉξ€·π‘₯,π‘€π‘šξ€Έξ€·π‘€π‘‘π‘₯⟢0,πΌπ‘šξ€Έ=4𝑐+π‘œπ‘š(1).(3.31) However, applying Lemma 2.4 with βˆšπ‘‘=2𝑐/β€–π‘’π‘šβ€–πΈ, we have πΌξ€·π‘€π‘šξ€Έβ‰€1+𝑑2𝑒2π‘š+πΌπ‘šξ€ΈβŸΆπ‘,(π‘šβŸΆβˆž),(3.32) which contradicts (3.31), thus (3.30) holds.
On the other hand, similar to case (i), (3.13) holds. Let Ω=Ω⧡{π‘₯βˆˆΞ©βˆΆπ‘€(π‘₯)=0}. Then |Ω|>0 by (3.30). From assumptions (𝑓3) and (𝑓4), π‘π‘š(π‘₯)β‰₯0 and π‘π‘š(π‘₯)β†’+∞ as π‘šβ†’βˆž in Ω, where π‘π‘š(π‘₯) is defined by (3.5). Hence, from (3.8) and (3.13), we have4𝑐=liminfπ‘šβ†’βˆžβ€–β€–π‘€π‘šβ€–β€–2𝐸=liminfπ‘šβ†’βˆžξ€œΞ©π‘π‘š(||𝑀π‘₯)π‘š||2𝑑π‘₯β‰₯liminfπ‘šβ†’βˆžξ€œξ‚Ξ©π‘π‘š||𝑀(π‘₯)π‘š||2β‰₯ξ€œπ‘‘π‘₯Ωliminfπ‘šβ†’βˆžπ‘π‘š||𝑀(π‘₯)π‘š||2𝑑π‘₯=+∞,(3.33) which is a contradiction, thus β€–π‘’π‘šβ€–πΈβ†›+∞, that is, up to a subsequence, {π‘’π‘š} is bounded in 𝐸.

Proof of Theorem 1.1. The proof of this theorem is divided in two steps.
Step 1. There exists 𝜌>0, 𝛼>0 such that 𝐼(𝑒)>0 in 𝐡𝜌(0) and 𝐼(𝑒)|πœ•π΅πœŒβ‰₯𝛼.
In fact, in each case, assumptions (𝑓1)–(𝑓3) imply that for any πœ€>0, there exists πΆπœ€>0 such that, for all π‘’βˆˆβ„, there holds||||𝑓(π‘₯,𝑒)β‰€πœ€|𝑒|+πΆπœ€|𝑒|π‘žβˆ’1,||||𝐹(π‘₯,𝑒)β‰€πœ€|𝑒|2+πΆπœ€|𝑒|π‘ž,(3.34) where π‘ž is the same as that in (1.2), from which, it is easy to see that there exists 𝜌>0, 𝛼>0 such that 𝐼(𝑒)>0 in 𝐡𝜌(0) and 𝐼(𝑒)|πœ•π΅πœŒβ‰₯𝛼.
Step 2. By the Symmetric Mountain Pass Lemma 2.3, to prove Theorem 1.1, it suffices to prove that for any π‘˜β‰₯1, there exists a π‘˜-dimensional subspace πΈπ‘˜ of 𝐸 and π‘…π‘˜>0 such that 𝐼(𝑒)≀0,βˆ€π‘’βˆˆπΈπ‘˜β§΅π΅π‘…π‘˜.(3.35) First, we prove (3.35) in the case β„“βˆˆ(Ξ›π‘˜,+∞). Since β„“>Ξ›π‘˜, there is πœ€>0 such that β„“βˆ’πœ€>Ξ›π‘˜. By the definition of Ξ›π‘˜, there exists a π‘˜-dimensional subspace πΈπ‘˜ of 𝐸 such that, for the above πœ€>0, there holds supπ‘’βˆˆπΈπ‘˜β§΅{0}Ξ¨(𝑒)Ξ¦(𝑒)β‰€Ξ›π‘˜+πœ€2πœ€<π‘™βˆ’2,(3.36) that is, supπ‘’βˆˆπΈπ‘˜β§΅{0}Ξ¦(𝑒)>1Ξ¨(𝑒)β„“βˆ’πœ€/2.(3.37) By assumption (𝑓3), we have lim|𝑒|β†’+∞𝐹(π‘₯,𝑒)|𝑒|2=β„“2.(3.38) Then, for the above πœ€>0, there exists 𝑀>0 large enough such that 𝐹(π‘₯,𝑒)|𝑒|2>12ξ‚€πœ€β„“βˆ’4,βˆ€|𝑒|>𝑀.(3.39) Therefore, if π‘’βˆˆπΈπ‘˜ with ‖𝑒‖𝐸=𝑅, by (3.39) and (3.37), we obtain 1𝐼(𝑒)=2𝑅2βˆ’ξ€œΞ©β‰€1𝐹(π‘₯,𝑒)𝑑π‘₯2𝑅2βˆ’ξ€œ|𝑒|>𝑀≀1𝐹(π‘₯,𝑒)𝑑π‘₯βˆ’πΆ(𝑀,Ξ©)2𝑅2βˆ’12ξ‚€πœ€β„“βˆ’4ξ‚ξ€œΞ©|𝑒|2=𝑅𝑑π‘₯βˆ’πΆ(𝑀,Ξ©)22ξƒ©ξ‚€πœ€1βˆ’β„“βˆ’4ξ‚ξ€œΞ©ξ‚΅|𝑒|𝑅2ξƒͺ≀𝑅𝑑π‘₯βˆ’πΆ(𝑀,Ξ©)22ξ‚€1βˆ’β„“βˆ’πœ€/4ξ‚β„“βˆ’πœ€/2βˆ’πΆ(𝑀,Ξ©)<0,(3.40) if 𝑅β‰₯π‘…π‘˜ and π‘…π‘˜>0 large enough.
If β„“=+∞, similar to (3.37), for any π‘˜β‰₯1, there exists πΈπ‘˜βŠ‚πΈ such that supπ‘’βˆˆπΈπ‘˜β§΅{0}Ξ¦(𝑒)>1Ξ¨(𝑒)Ξ›π‘˜+1/2,(3.41) similar to (3.39), from assumption (𝑓3) with β„“=+∞ it follows that there exists π‘€π‘˜>0 such that 2𝐹(π‘₯,𝑒)|𝑒|2>Ξ›π‘˜+1,βˆ€|𝑒|>π‘€π‘˜.(3.42) Then, if π‘’βˆˆπΈπ‘˜ with ‖𝑒‖𝐸=𝑅, we have 𝑅𝐼(𝑒)≀22ξ‚΅Ξ›1βˆ’π‘˜+1Ξ›π‘˜ξ‚Άξ€·π‘€+1/2βˆ’πΆπ‘˜ξ€Έ,π‘˜,Ξ©<0,(3.43) if 𝑅β‰₯π‘…π‘˜ and π‘…π‘˜>0 large enough. This completes the proof of Theorem 1.1.

Acknowledgment

The authors would like to thank the referees for carefully reading this paper and making valuable comments and suggestions.

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