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

Abstract

We study the following fourth-order elliptic equations: Δ²𝑢+𝑎Δ𝑢=𝑓(𝑥,𝑢),𝑥∈Ω,𝑢=Δ𝑢=0,𝑥∈𝜕Ω, where Ω⊂ℝ^𝑁 is a bounded domain with smooth boundary 𝜕Ω and 𝑓(𝑥,𝑢) is asymptotically linear with respect to 𝑢 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: Δ²𝑢𝑢+=𝑎ΔΔ𝑢𝑢==𝑓0,(𝑥,𝑥𝑢)∈,𝜕Ω𝑥,∈Ω,(1.1) where Ω⊂ℝ^𝑁 is a bounded domain with smooth boundary 𝜕Ω, Δ² is the biharmonic operator, 𝑎<𝜆₁ (𝜆₁ is the first eigenvalue of −Δ in 𝐻¹⁰(Ω)) is a parameter. We assume that 𝑓(𝑥,𝑢) satisfies the following hypotheses.(𝑓₁)𝑓(𝑥,𝑢)∈𝐶(Ω×ℝ,ℝ).(𝑓₂)lim_|𝑢|→0𝑓(𝑥,𝑢)/𝑢=0 uniformly for 𝑥∈Ω.(𝑓₃)lim_|𝑢|→∞𝑓(𝑥,𝑢)/𝑢=ℓ uniformly for 𝑥∈Ω, where ℓ∈(0,+∞) is a constant, or ℓ=+∞, and there exists 𝐶>0, 𝑞∈[2,2^∗) such that ||𝑓(𝑥,𝑢)||≤𝐶1+|𝑢|^𝑞−1,(1.2)where 2^∗=2𝑁/(𝑁−4).(𝑓₄)𝑓(𝑥,𝑢) is odd in 𝑢.(𝑓₅)lim_|𝑢|→∞(𝑓(𝑥,𝑢)𝑢−2𝐹(𝑥,𝑢))=+∞ uniformly for 𝑥∈Ω, where𝐹(𝑥,𝑢)=∫^𝑢0𝑓(𝑥,𝑡)𝑑𝑡.(𝑓₆)𝑓(𝑥,𝑢)/𝑢 is nondecreasing with respect to 𝑢≥0, for a.e. 𝑥∈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: Δ²𝑢+𝑎Δ𝑢𝑢==Δ𝑑𝑢=(𝑢0+,1)𝑥⁺∈−𝜕1Ω,,𝑥∈Ω,(1.3) where 𝑢⁺=max{𝑢,0} and 𝑑∈ℝ. 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 2𝑘−1 solutions when 𝑁=1 and 𝑑>𝜆_𝑘(𝜆_𝑘−𝑐) (𝜆_𝑘 is the sequence of the eigenvalues of −Δ in 𝐻¹⁰(Ω)) by the global bifurcation method. In [4], Tarantello found a negative solution when 𝑑>𝜆₁(𝜆₁−𝑐) by a degree argument. For Problem (1.1), when 𝑓(𝑥,𝑢)=𝑏𝑔(𝑥,𝑢), the existence of two or three nontrivial solutions has been obtained in [5, 6] for 𝑔(𝑥,𝑢) 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(𝑥,𝑢) is asymptotically linear with respect to 𝑢 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(Ω)∩𝐻¹⁰(Ω), then 𝐸 is a Hilbert space with the following inner product and the norm: ⟨𝑢,𝑣⟩_𝐸=_Ω(Δ𝑢Δ𝑣−𝑎∇𝑢∇𝑣)𝑑𝑥,‖𝑢‖_𝐸=⟨𝑢,𝑢⟩^𝐸1/2.(1.4)

The corresponding energy functional of problem (1.1) is defined on 𝐸 by 𝐼(𝑢)=12_Ω||Δ𝑢||²−𝑎||∇𝑢||²𝑑𝑥−_Ω𝐹(𝑥,𝑢)𝑑𝑥,(1.5) where 𝐹(𝑥,𝑢)=∫^𝑢0𝑓(𝑥,𝑡)𝑑𝑡. From (𝑓₁)–(𝑓₃), it is easy to see that 𝐼∈𝐶¹(𝐸,ℝ), it is well known that the weak solutions of problem (1.1) are the critical points of the energy functional 𝐼(𝑢).

Our main results are stated as follows.

Theorem 1.1. Assume that 𝑓(𝑥,𝑢) satisfies assumptions (𝑓₁)–(𝑓₄), and Λ_𝑘 is given by (2.8). Then the following hold. (i)If ℓ∈(Λ_𝑘,+∞) is not an eigenvalue of problem (2.4), then problem (1.1) has at least 𝑘 pairs of nontrivial solutions in 𝐸.(ii)Suppose that (𝑓₅) is satisfied, then the conclusion of (i) holds even if ℓ is an eigenvalue of problem (2.4).(iii)If ℓ=+∞, 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 |Ω| the Lebesgue measure of Ω, 𝐵_𝜌={𝑢∈𝐸∶‖𝑢‖_𝐸<𝜌}. 𝐶 will denote various positive constants, → (respectively⇀) denotes strong (respectively weak) convergence. 𝑜_𝑚(1) denote 𝑜_𝑚(1)→0 as 𝑚→∞. 𝐿^𝑠(Ω),(1≤𝑠<+∞) denote Lebesgue spaces, the norm 𝐿^𝑠 is denoted by |⋅|_𝑠 for 1≤𝑠<+∞. The dual space of a Banach space 𝐸 will be denoted by 𝐸⁻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∈𝐶¹(𝐸,ℝ) is said to satisfy condition (𝐶) at level 𝑐∈ℝ ((𝐶)_𝑐 for short), if the following fact is true: any sequence {𝑢_𝑚}⊂𝐸, which satisfies𝐼𝑢_𝑚⟶𝑐,1+‖𝑢_𝑚‖_𝐸‖𝐼^𝑢_𝑚‖_𝐸⁻¹⟶0,(𝑚⟶∞)(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 𝑢∈𝐴 implies −𝑢∈𝐴. For a closed symmetric set 𝐴 which does not contain the origin, we define a genus 𝛾(𝐴) of 𝐴 by the smallest integer 𝑘 such that there exists an odd continuous mapping from 𝐴 to ℝ^𝑘⧵{0}. If there does not exist such a 𝑘, we define 𝛾(𝐴)=∞. Moreover, we set 𝛾(∅)=0.

Let 𝐸 be an infinite dimensional real Banach space,  𝐼∈𝐶¹(𝐸,ℝ), 𝐴₀={𝑢∈𝐸∶𝐼(𝑢)≥0}, Γ^∗={ℎ(0)=0,ℎisanoddhomeomorphismof𝐸andℎ(𝐵₁)⊂𝐴₀}, Γ_𝑚={𝒦⊂𝐸∶𝒦 is compact, symmetric with respect to the origin, and for any ℎ∈Γ^∗, there holds 𝛾(𝒦∩ℎ(𝜕𝐵₁))≥𝑚}. If Γ_𝑚≠∅, define𝑏_𝑚=inf_{𝒦⊂Γ𝑚}max_𝑢∈𝒦𝐼(𝑢).(2.2)

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

Lemma 2.3. Let 𝑒₁,𝑒₂,…,𝑒_𝑚,… be linearly independent in 𝐸, and 𝐸_𝑖=span{𝑒₁,𝑒₂,…,𝑒_𝑖}, 𝑖=1,2,…,𝑚,…. Suppose that 𝐼∈𝐶¹(𝐸,ℝ) satisfies 𝐼(0)=0, 𝐼(−𝑢)=𝐼(𝑢), and (𝐶)_𝑐 condition for 𝑐≥0. Furthermore, there exists 𝜌>0, 𝛼>0 such that 𝐼(𝑢)>0 in 𝐵_𝜌⧵{0} and 𝐼(𝑢)|_𝜕𝐵𝜌≥𝛼. Then, if 𝐸_𝑚∩𝐴₀ is bounded, then Γ_𝑚≠∅ and 𝑏_𝑚≥𝛼>0 is a critical value of 𝐼. Moreover, if 𝐸_𝑚+𝑖∩𝐴₀ is bounded for all 𝑖=1,…𝑟, and𝑏_𝑚+1=⋯=𝑏_𝑚+𝑟=𝑏,(2.3) then 𝛾(𝐾_𝑏)≥𝑟, where 𝐾_𝑏={𝑢∈𝐸∶𝐼(𝑢)=𝑏,𝐼^(𝑢)=0}. If 𝐸_𝑚∩𝐴₀ is bounded for all 𝑚, then 𝐼(𝑢) possesses infinitely many critical values.

Let us consider the eigenvalue problem:Δ²𝑢𝑢=+Δ𝑎Δ𝑢𝑢==0,Λ𝑢𝑥,∈𝑥𝜕∈Ω.Ω,(2.4)Set Φ(𝑢)=_Ω||Δ𝑢||²−𝑎||∇𝑢||²𝑑𝑥,Ψ(𝑢)=_Ω|𝑢|²𝑑𝑥.(2.5)For 𝑎<𝜆₁, Φ(𝑢) and Ψ(𝑢) are well defined. Furthermore, Φ(𝑢),Ψ(𝑢)∈𝐶¹(𝐸,ℝ), and a real value Λ is an eigenvalue of problem (2.4) if and only if there exists 𝑢∈𝐸⧵{0} such that Φ^(𝑢)=ΛΨ^(𝑢). At this point, let us set 𝒩={𝑢∈𝐸∶Ψ(𝑢)=1}.(2.6)Then 𝒩≠∅ and 𝒩 is a 𝐶¹ manifold in 𝐸. It follows from the standard Lagrange multiples arguments that eigenvalues of (2.4) correspond to critical values of Φ|_𝒩, and Φ satisfies the (PS) condition on 𝒩. Thus a sequence of critical values of Φ|_𝒩 comes from the Ljusternik-Schnirelmann critical point theory on 𝐶¹ manifolds. For any 𝑘∈𝑁, set Γ_𝑘={𝐴⊂𝒩∶𝐴iscompact,symmetricand𝛾(𝐴)≥𝑘}.(2.7) Then values: Λ_𝑘∶=inf_{𝐴∈Γ𝑘}max_𝑢∈𝐴Φ(𝑢)(2.8)are critical values and hence are eigenvalues of problem (2.4). Moreover, 0<Λ₁<Λ₂≤Λ₃≤⋯≤Λ_𝑘≤⋯→+∞.

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

Lemma 2.4. For the functional 𝐼(𝑢) defined by (1.5), if assumptions (𝑓₁) and (𝑓₆) hold, and for any {𝑢_𝑚}⊂𝐸 with ⟨𝐼^(𝑢_𝑚),𝑢_𝑚⟩→0 as 𝑚→∞, then there is a subsequence, still denoted by {𝑢_𝑚}, such that𝐼𝑡𝑢_𝑚≤1+𝑡²2𝑚+𝐼𝑢^𝑚(2.9) holds for all 𝑡>0, 𝑚∈𝑁⁺.

Proof. This lemma is essentially due to [27, 28]. For the sake of completeness, we prove it here.
By ⟨𝐼′(𝑢_𝑚),𝑢_𝑚⟩→0 as 𝑚→∞, for a suitable subsequence, we may assume that −1𝑚<𝐼′𝑢^𝑚,𝑢_𝑚=‖‖𝑢_𝑚‖‖^2𝐸−_Ω𝑓𝑥,𝑢_𝑚𝑢_𝑚𝑑𝑥<1𝑚,∀𝑚.(2.10) We claim that for any 𝑡>0 and 𝑚∈𝑁⁺, 𝐼𝑡𝑢_𝑚<𝑡²2𝑚+_Ω21𝑓𝑥,𝑢^𝑚𝑢_𝑚−𝐹𝑥,𝑢_𝑚𝑑𝑥.(2.11) Indeed, for any 𝑡>0, at fixed 𝑥∈Ω and 𝑚∈𝑁⁺, we set ℎ(𝑡)=𝑡²2𝑓𝑥,𝑢^𝑚𝑢_𝑚−𝐹𝑥,𝑡𝑢_𝑚,(2.12) then ℎ^(𝑡)=𝑡𝑓𝑥,𝑢_𝑚𝑢_𝑚−𝑓𝑥,𝑡𝑢_𝑚𝑢_𝑚=𝑡𝑢_𝑚𝑓𝑥,𝑢^𝑚−1𝑡𝑓𝑥,𝑡𝑢^𝑚≥≤00,,0<𝑡≥𝑡≤1,1,by𝑓₆,(2.13) hence ℎ(𝑡)≤ℎ(1)∀𝑡>0.(2.14) Therefore, 𝐼𝑡𝑢_𝑚=𝑡²2‖‖𝑢^𝑚≤𝑡‖‖^2𝐸−_Ω𝐹𝑥,𝑡𝑢_𝑚𝑑𝑥₂2𝑚1+_Ω≤𝑡𝑓𝑥,𝑢_𝑚𝑢_𝑚𝑑𝑥−_Ω𝐹𝑥,𝑡𝑢_𝑚𝑑𝑥by(2.10)₂2𝑚+_Ω𝑡2²𝑓𝑥,𝑢^𝑚𝑢_𝑚≤𝑡−𝐹𝑥,𝑡𝑢_𝑚𝑑𝑥₂2𝑚+_Ω21𝑓𝑥,𝑢^𝑚𝑢_𝑚−𝐹𝑥,𝑢_𝑚𝑑𝑥by(2.14)(2.15)and our claim (2.11) is proved.
On the other hand, 𝐼𝑢_𝑚=12‖‖𝑢^𝑚≥1‖‖^2𝐸−_Ω𝐹𝑥,𝑢_𝑚𝑑𝑥2−𝑚1+_Ω𝑓𝑥,𝑢_𝑚𝑢_𝑚𝑑𝑥−_Ω𝐹𝑥,𝑢_𝑚𝑑𝑥,(2.16) that is, _Ω21𝑓𝑥,𝑢^𝑚𝑢_𝑚−𝐹𝑥,𝑢_𝑚𝑑𝑥≤12𝑚+𝐼𝑢^𝑚.(2.17) Combining (2.11) and (2.17) we have that 𝐼𝑡𝑢_𝑚≤1+𝑡²2𝑚+𝐼𝑢^𝑚,∀𝑡>0,𝑚∈𝑁⁺.(2.18)
The proof is completed.

3. Proof of the Main Results

We begin with the following lemma.

Lemma 3.1. Let 𝑐≥0. Assume that 𝑓(𝑥,𝑢) satisfies assumptions (𝑓₁)–(𝑓₃). Then the following hold. (i)𝐼(𝑢) satisfies (𝐶)_𝑐 condition if ℓ<+∞ in assumption (𝑓₃), and ℓ is not an eigenvalue of problem (2.4).(ii)If ℓ<+∞ is an eigenvalue of problem (2.4) and (𝑓₅) holds, then 𝐼(𝑢) satisfies (𝐶)_𝑐 condition.(iii)If ℓ=+∞, and (𝑓₆) holds, then 𝐼(𝑢) satisfies (𝐶)_𝑐 condition.

Proof. Suppose that {𝑢_𝑚}⊂𝐸 is a (𝐶)_𝑐 sequence, that is, as 𝑚→∞, we have 1+‖‖𝑢_𝑚𝐼𝑢_𝑚⟶𝑐≥0,(3.1)‖‖_𝐸‖‖𝐼′(𝑢_𝑚)‖‖_𝐸−1⟶0,in𝐸⁻¹.(3.2) It is easy to see that (3.2) implies that as 𝑚→∞, there hold ‖‖𝑢_𝑚‖‖^2𝐸−_Ω𝑓𝑥,𝑢_𝑚𝑢_𝑚𝑑𝑥=𝑜_𝑚(1),(3.3)_ΩΔ𝑢_𝑚Δ𝜑−𝑎∇𝑢_𝑚∇𝜑𝑑𝑥−_Ω𝑓𝑥,𝑢_𝑚𝜑𝑑𝑥=𝑜_𝑚(1),∀𝜑∈𝐸.(3.4) By Sobolev compact embedding, to show that 𝐼(𝑢) satisfies (𝐶)_𝑐 condition, it suffices to show the boundedness of (𝐶)_𝑐 sequence in 𝐸 for each case.
(i) Suppose that 0<ℓ<+∞ and ℓ is not an eigenvalue of problem (2.4). Arguing by contradiction, we suppose that there exists a subsequence, still denoted by {𝑢_𝑚}, such that as 𝑚→∞, there holds ‖𝑢_𝑚‖_𝐸→+∞. Define 𝑝_𝑚(𝑥)=⎧⎪⎨⎪⎩𝑓𝑥,𝑢^𝑚(𝑥)𝑢_𝑚(𝑥),𝑢^𝑚0,𝑢_𝑚(𝑥)≠0,(𝑥)=0.(3.5) Then from assumptions (𝑓₁)–(𝑓₃), there exists 𝑀>0 such that 0≤𝑝_𝑚(𝑥)≤𝑀.(3.6) Let 𝑤_𝑚=𝑢^𝑚‖𝑢_𝑚‖_𝐸.(3.7) Obviously, 𝑤_𝑚 is bounded in 𝐸. Going if necessary to a subsequence, we can assume that 𝑤𝑤_𝑚⇀𝑤,weaklyin𝐸,_𝑚𝑤⟶𝑤,a.e.inΩ,_𝑚⟶𝑤,stronglyin𝐿^𝑠(Ω),∀𝑠∈2,2^∗.(3.8) It is easy to show that 𝑤≢0. In fact, if 𝑤≡0, then from (3.3), (3.6), (3.8) and the definitions of 𝑝_𝑚 and 𝑤_𝑚, as 𝑚→∞, we have 1=‖‖𝑤_𝑚‖‖^2𝐸=_Ω𝑝_𝑚(𝑥)||𝑤_𝑚||²𝑑𝑥+𝑜_𝑚(1)≤𝑀_Ω||𝑤_𝑚||²𝑑𝑥+𝑜_𝑚(1)⟶0,(3.9) which is a contradiction.
From (3.6), there exists ℎ(𝑥)∈𝐿^∞(Ω) with 0≤ℎ(𝑥)≤𝑀 such that, up to a subsequence, as 𝑚→∞, there holds 𝑝_𝑚(𝑥)⇀ℎ(𝑥),weakly^∗in𝐿^∞(Ω).(3.10) Then from (3.8) it follows that 𝑝_𝑚(𝑥)𝑤_𝑚⇀ℎ(𝑥)𝑤weaklyin𝐿²(Ω),_Ω𝑝_𝑚(𝑥)||𝑤_𝑚||²𝑑𝑥⟶_Ωℎ(𝑥)|𝑤|²𝑑𝑥.(3.11) On the other hand, from (3.3), (3.4), (3.5), and (3.7), we have _ΩΔ𝑤_𝑚Δ𝜑−𝑎∇𝑤_𝑚∇𝜑𝑑𝑥=_Ω𝑝_𝑚(𝑥)𝑤_𝑚𝜑𝑑𝑥+𝑜_𝑚(1),∀𝜑∈𝐸.(3.12)_Ω||Δ𝑤^𝑚||²−𝑎||∇𝑤_𝑚||²𝑑𝑥=_Ω𝑝_𝑚(𝑥)||𝑤_𝑚||²𝑑𝑥+𝑜_𝑚(1).(3.13) It follows from (3.11)–(3.13) that ‖‖𝑤_𝑚‖‖^2𝐸=_Ωℎ(𝑥)|𝑤|²𝑑𝑥+𝑜_𝑚(1),_ΩΔ𝑤_𝑚Δ𝜑−𝑎∇𝑤_𝑚∇𝜑𝑑𝑥=_Ωℎ(𝑥)𝑤𝜑𝑑𝑥+𝑜_𝑚(1),∀𝜑∈𝐸.(3.14) Therefore (3.14) implies that 𝑤 satisfies _Ω(Δ𝑤Δ𝜑−𝑎∇𝑤∇𝜑)𝑑𝑥=_Ωℎ(𝑥)𝑤𝜑𝑑𝑥,∀𝜑∈𝐸.(3.15) Let ΩΩ⁰={𝑥∈Ω∶𝑤(𝑥)=0},₊Ω={𝑥∈Ω∶𝑤(𝑥)>0},₋={𝑥∈Ω∶𝑤(𝑥)<0}.(3.16) Then 𝑢_𝑚(𝑥)→+∞ as 𝑚→∞ if 𝑥∈Ω⁺, and 𝑢_𝑚(𝑥)→−∞ as 𝑚→∞ if 𝑥∈Ω⁻. From assumption (𝑓₃), ℎ(𝑥)≡ℓ for all 𝑥∈Ω⁺∪Ω⁻. Thus (3.15) implies that 𝑤 satisfies _Ω0(Δ𝑤Δ𝜑−𝑎∇𝑤∇𝜑−ℎ(𝑥)𝑤𝜑)𝑑𝑥+_Ω+∪Ω−(Δ𝑤Δ𝜑−𝑎∇𝑤∇𝜑−ℎ(𝑥)𝑤𝜑)𝑑𝑥=0,∀𝜑∈𝐸.(3.17) Therefore _Ω(Δ𝑤Δ𝜑−𝑎∇𝑤∇𝜑)𝑑𝑥=ℓ_Ω𝑤𝜑𝑑𝑥,∀𝜑∈𝐸.(3.18) This means that ℓ is an eigenvalue of problem (2.4), which contradicts our assumption, so {𝑢_𝑚} is bounded in 𝐸.
(ii) Suppose ℓ∈(0,+∞) is an eigenvalue of problem (2.4), we need the additional assumption (𝑓₅).
From assumption (𝑓₅), there exists 𝑇₀>0 such that𝑓(𝑥,𝑢)𝑢−2𝐹(𝑥,𝑢)≥0,∀|𝑢|≥𝑇₀,𝑥∈Ω,(3.19) and there exists 𝐶₀=𝐶₀(𝑇₀)>0 such that _{{|𝑢𝑚|≤𝑇0}}𝑓𝑥,𝑢_𝑚𝑢_𝑚−2𝐹𝑥,𝑢_𝑚𝑑𝑥≥−𝐶₀.(3.20) Furthermore, under assumptions (𝑓₁)–(𝑓₃), there exists 𝑀>0 such that ||𝑓(𝑥,𝑢)||≤𝑀|𝑢|,||𝐹(𝑥,𝑢)||≤𝑀2|𝑢|²,∀𝑥∈Ω.(3.21) Let 𝐾=(2𝑐+𝐶₀)(2𝑀𝑆)^𝑁/2, where 𝑀>0 is given by (3.21), 𝑆>0 is the best Sobolev constant such that _Ω|𝑢|^2∗𝑑𝑥^2/2∗≤𝑆_Ω||Δ𝑢||²−𝑎||∇𝑢||²𝑑𝑥,∀𝑢∈𝐸.(3.22) From assumption (𝑓₅), there exists 𝑇=𝑇(𝐾)>𝑇₀>0 such that 𝑓(𝑥,𝑢)𝑢−2𝐹(𝑥,𝑢)≥𝐾,∀|𝑢|≥𝑇,𝑥∈Ω.(3.23) For the above 𝑇>0 and each 𝑚≥1, set 𝐴_𝑚=𝑥∈Ω∶||𝑢_𝑚(𝑥)||≥𝑇,𝐵_𝑚=𝑥∈Ω∶||𝑢_𝑚(𝑥)||≤𝑇.(3.24) From estimates (3.20), (3.1), (3.3), and (3.23), we get 2𝑐+𝑜_𝑚(1)=≥_Ω𝑓𝑥,𝑢_𝑚𝑢_𝑚−2𝐹𝑥,𝑢_𝑚𝑑𝑥_𝐴𝑚𝑓𝑥,𝑢_𝑚𝑢_𝑚−2𝐹𝑥,𝑢_𝑚𝑑𝑥−𝐶₀≥𝐾||𝐴_𝑚||−𝐶₀,(3.25) where |𝐴_𝑚| denotes the measure of 𝐴_𝑚.
On the other hand, for any fixed 𝑟>2, from (3.1) and (3.3), we have21−1𝑟‖‖𝑢^𝑚‖‖^2𝐸−_Ω𝐹𝑥,𝑢^𝑚−1𝑟𝑓𝑥,𝑢^𝑚𝑢_𝑚𝑑𝑥=𝑐+𝑜^𝑚(1).(3.26) Since Ω is bounded and 𝑓∈𝐶(Ω×ℝ,ℝ), there exists a constant 𝐶=𝐶(Ω,𝑓,𝑇) such that ||||_𝐵𝑚𝐹𝑥,𝑢^𝑚−1𝑟𝑓𝑥,𝑢^𝑚𝑢_𝑚𝑑𝑥||||≤𝐶,∀𝑥∈Ω.(3.27) Then, from (3.21)–(3.26), Hölder inequality and Sobolev inequality, we have 𝑐+𝑜_𝑚(1)≥12−1𝑟‖‖𝑢^𝑚‖‖^2𝐸−𝐶−_𝐴𝑚𝐹𝑥,𝑢^𝑚−1≥1𝑟𝑓𝑥,𝑢^𝑚𝑢_𝑚𝑑𝑥2−1𝑟‖‖𝑢^𝑚‖‖^2𝐸−𝐶−_𝐴𝑚12𝑓𝑥,𝑢^𝑚𝑢_𝑚−1=1𝑟𝑓𝑥,𝑢^𝑚𝑢_𝑚𝑑𝑥2−1𝑟‖‖𝑢^𝑚‖‖^2𝐸−𝐶−12−1𝑟_𝐴𝑚≥1𝑓𝑥,𝑢_𝑚𝑢_𝑚𝑑𝑥2−1𝑟‖‖𝑢^𝑚‖‖^2𝐸−𝐶−12−1𝑟𝑀_𝐴𝑚≥1||𝑢_𝑚||²𝑑𝑥2−1𝑟‖‖𝑢^𝑚‖‖^2𝐸−𝐶−12−1𝑟𝑀||𝑢^𝑚||²²_∗||𝐴_𝑚||^2/𝑁≥12−1𝑟‖‖𝑢^𝑚‖‖^2𝐸−𝐶−12−1𝑟𝑀𝑆‖‖𝑢^𝑚‖‖^2𝐸||||2𝑐𝐾+𝐶⁰||||+𝑜^𝑚(1)^2/𝑁≥1212−1𝑟‖‖𝑢^𝑚‖‖^2𝐸−𝐶−12−1𝑟𝑀𝑆‖‖𝑢^𝑚‖‖^2𝐸⋅𝑜_𝑚(1),(3.28) that is, {𝑢_𝑚} is bounded in 𝐸.
(iii) Finally, we prove the case ℓ=+∞. 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 𝑡_𝑚=2√𝑐‖‖𝑢_𝑚‖‖_𝐸,𝑤_𝑚=𝑡_𝑚𝑢_𝑚=2√𝑐𝑢_𝑚‖‖𝑢_𝑚‖‖_𝐸.(3.29) Then ‖𝑤_𝑚‖_𝐸=2√𝑐 and {𝑤_𝑚} is bounded in 𝐸. Hence, up to a subsequence, we may assume that: there exists 𝑤∈𝐸 such that (3.8) also holds in this case. If ‖𝑢_𝑚‖_𝐸→+∞, we claim that 𝑤(𝑥)≢0.(3.30) In fact, if 𝑤(𝑥)≡0 in Ω, then (3.29) and (3.8) imply that _Ω𝐹𝑥,𝑤_𝑚𝑑𝑥⟶0,𝐼𝑤_𝑚=4𝑐+𝑜_𝑚(1).(3.31) However, applying Lemma 2.4 with 𝑡=2√𝑐/‖𝑢_𝑚‖_𝐸, we have 𝐼𝑤_𝑚≤1+𝑡²2𝑚+𝐼𝑢^𝑚⟶𝑐,(𝑚⟶∞),(3.32) which contradicts (3.31), thus (3.30) holds.
On the other hand, similar to case (i), (3.13) holds. Let Ω=Ω⧵{𝑥∈Ω∶𝑤(𝑥)=0}. Then |Ω|>0 by (3.30). From assumptions (𝑓₃) and (𝑓₄), 𝑝_𝑚(𝑥)≥0 and 𝑝_𝑚(𝑥)→+∞ as 𝑚→∞ in Ω, where 𝑝_𝑚(𝑥) is defined by (3.5). Hence, from (3.8) and (3.13), we have4𝑐=liminf_𝑚→∞‖‖𝑤_𝑚‖‖^2𝐸=liminf_𝑚→∞_Ω≥liminf_𝑚→∞𝑝_𝑚(𝑥)||𝑤_𝑚||²𝑑𝑥_Ω≥𝑝_𝑚(𝑥)||𝑤_𝑚||²𝑑𝑥_Ωliminf_𝑚→∞𝑝_𝑚(𝑥)||𝑤_𝑚||²𝑑𝑥=+∞,(3.33) which is a contradiction, thus ‖𝑢_𝑚‖_𝐸↛+∞, that is, up to a subsequence, {𝑢_𝑚} 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(𝑢)>0 in 𝐵_𝜌(0) and 𝐼(𝑢)|_𝜕𝐵𝜌≥𝛼.
In fact, in each case, assumptions (𝑓₁)–(𝑓₃) imply that for any 𝜀>0, there exists 𝐶_𝜀>0 such that, for all 𝑢∈ℝ, there holds||𝑓(𝑥,𝑢)||≤𝜀|𝑢|+𝐶_𝜀|𝑢|^𝑞−1,||𝐹(𝑥,𝑢)||≤𝜀|𝑢|²+𝐶_𝜀|𝑢|^𝑞,(3.34) where 𝑞 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(𝑢)>0 in 𝐵_𝜌(0) and 𝐼(𝑢)|_𝜕𝐵𝜌≥𝛼.Step 2. By the Symmetric Mountain Pass Lemma 2.3, to prove Theorem 1.1, it suffices to prove that for any 𝑘≥1, there exists a 𝑘-dimensional subspace 𝐸_𝑘 of 𝐸 and 𝑅_𝑘>0 such that 𝐼(𝑢)≤0,∀𝑢∈𝐸_𝑘⧵𝐵_𝑅𝑘.(3.35) First, we prove (3.35) in the case ℓ∈(Λ_𝑘,+∞). Since ℓ>Λ_𝑘, there is 𝜀>0 such that ℓ−𝜀>Λ_𝑘. By the definition of Λ_𝑘, there exists a 𝑘-dimensional subspace 𝐸_𝑘 of 𝐸 such that, for the above 𝜀>0, there holds sup_{𝑢∈𝐸𝑘⧵{0}}Ψ(𝑢)Φ(𝑢)≤Λ^𝑘+𝜀2<𝑙−2𝜀,(3.36) that is, sup_{𝑢∈𝐸𝑘⧵{0}}Φ(𝑢)Ψ(𝑢)>ℓ−1𝜀/2.(3.37) By assumption (𝑓₃), we have lim_{|𝑢|→+∞}𝐹(𝑥,𝑢)|𝑢|²=ℓ2.(3.38) Then, for the above 𝜀>0, there exists 𝑀>0 large enough such that 𝐹(𝑥,𝑢)|𝑢|²>12ℓ−4𝜀,∀|𝑢|>𝑀.(3.39) Therefore, if 𝑢∈𝐸_𝑘 with ‖𝑢‖_𝐸=𝑅, by (3.39) and (3.37), we obtain 𝐼(𝑢)=12𝑅²≤1−_Ω𝐹(𝑥,𝑢)𝑑𝑥2𝑅²≤1−_|𝑢|>𝑀𝐹(𝑥,𝑢)𝑑𝑥−𝐶(𝑀,Ω)2𝑅²−12ℓ−4𝜀_Ω=𝑅|𝑢|²𝑑𝑥−𝐶(𝑀,Ω)₂21−ℓ−4𝜀_Ω|𝑅𝑢|²≤𝑅𝑑𝑥−𝐶(𝑀,Ω)₂<02,1−ℓℓ−−𝜀𝜀//42−𝐶(𝑀,Ω)(3.40) if 𝑅≥𝑅_𝑘 and 𝑅_𝑘>0 large enough.
If ℓ=+∞, similar to (3.37), for any 𝑘≥1, there exists 𝐸_𝑘⊂𝐸 such that sup_{𝑢∈𝐸𝑘⧵{0}}Φ(𝑢)Ψ(𝑢)>Λ_𝑘+11/2,(3.41) similar to (3.39), from assumption (𝑓₃) with ℓ=+∞ it follows that there exists 𝑀_𝑘>0 such that 2𝐹(𝑥,𝑢)|𝑢|²>Λ_𝑘+1,∀|𝑢|>𝑀_𝑘.(3.42) Then, if 𝑢∈𝐸_𝑘 with ‖𝑢‖_𝐸=𝑅, we have 𝐼(𝑢)≤𝑅²21−Λ^𝑘+1Λ_𝑘+1/2−𝐶𝑀^𝑘,𝑘,Ω<0,(3.43) if 𝑅≥𝑅_𝑘 and 𝑅_𝑘>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.