Existence and Multiplicity of Nontrivial Solutions for a Class of Fourth-Order Elliptic Equations

Abstract

Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.

1. Introduction and Main Results

Consider the fourth-order Navier boundary value problem Δ²𝑢+𝑐Δ𝑢+𝑎(𝑥)𝑢=𝑓(𝑥,𝑢),inΩ,𝑢=Δ𝑢=0,on𝜕Ω, (1) where Ω⊂ℝ^𝑁(𝑁>4) is a bounded smooth domain, 𝑎∈𝐿^∞(Ω),𝑐∈ℝ, and 𝑓∈𝐶(Ω×ℝ,ℝ). Δ is the Laplace operator and Δ² is the biharmonic operator.

Let 0<𝜆₁<𝜆₂≤𝜆₃≤⋯≤𝜆_𝑘≤⋯ be the eigenvalues of −Δ in 𝐻¹₀(Ω). The eigenvalue problem Δ²𝑢+𝑐Δ𝑢=𝜇𝑢,inΩ,𝑢=Δ𝑢=0,on𝜕Ω, (2) has infinitely many eigenvalues 𝜇_𝑖=𝜆_𝑖(𝜆_𝑖−𝑐),𝑖=1,2,….

We will always assume 𝑐<𝜆₁. Let 𝐸 be the Hilbert space 𝐻²(Ω)⋂𝐻¹₀(Ω).  𝐸 is equipped with the inner product ⟨𝑢,𝑣⟩=_Ω(Δ𝑢Δ𝑣−𝑐∇𝑢∇𝑣)𝑑𝑥 (3) and the norm ‖𝑢‖=_Ω|Δ𝑢|²−𝑐|∇𝑢|²𝑑𝑥^1/2. (4)

A weak solution of problem (1) is any 𝑢∈𝐸 such that _Ω(Δ𝑢Δ𝑣−𝑐∇𝑢⋅∇𝑣+𝑎(𝑥)𝑢𝑣)𝑑𝑥−_Ω𝑓(𝑥,𝑢)𝑣𝑑𝑥=0 (5) for any 𝑣∈𝐸.

Let 𝜑∶𝐸→ℝ be the functional defined by 𝜑(𝑢)=12_Ω|Δ𝑢|²−𝑐|∇𝑢|²+𝑎(𝑥)𝑢²𝑑𝑥−_Ω𝐹(𝑥,𝑢)𝑑𝑥, (6) where 𝐹(𝑥,𝑢)=∫^𝑢₀𝑓(𝑥,𝑠)𝑑𝑠. And, one has 𝜑^′(𝑢),𝑣=_Ω((𝑥))−Δ𝑢Δ𝑣−𝑐∇𝑢⋅∇𝑣+𝑎𝑢𝑣𝑑𝑥_Ω𝑓(𝑥,𝑢)𝑣𝑑𝑥 (7) for any 𝑢,𝑣∈𝐸, so that a critical point of the functional 𝜑 in 𝐸 corresponds to a weak solution of problem (1).

In recent years, fourth-order problems have been studied by many authors. In [1], Lazer and McKenna have pointed out that problem (1) furnishes a model to study travelling waves in suspension bridges if 𝑓(𝑥,𝑢)=𝑏((𝑢+1)⁺−1), where 𝑢⁺=max{𝑢,0} and 𝑏∈ℝ. Since then, more general nonlinear fourth-order elliptic boundary value problems have been studied.

In [2, 3], Micheletti and Pistoia proved that problem Δ²𝑢+𝑐Δ𝑢=𝑓(𝑥,𝑢),inΩ,𝑢=Δ𝑢=0,onΩ, (8) admits two or three solutions by variational method. In [4], Zhang obtained the existence of weak solutions for problem (8) when 𝑓(𝑥,𝑢) is sublinear at ∞. In [5], Zhang and Li showed that problem (8) has at least two nontrivial solutions by means of Morse theory and local linking. When 𝑓(𝑥,𝑢) is asymptotically linear at infinity, the existence of three nontrivial solutions has been obtained in [6] by using Morse theory. In [7], by using the mountain pass theorem, An and Liu gave the existence result for nontrivial solutions for a class of asymptotically linear fourth-order elliptic equations. In [8], Zhou and Wu got the existence of four sign-changing solutions or infinitely many sign-changing solutions for (8) by using the sign-changing critical point theorems. In [9], Yang and Zhang showed new results on invariant sets of the gradient flows of the corresponding variational functionals and proved the existence of positive, negative, and sign-changing solutions for some fourth-order semilinear elliptic boundary value problems. In [10], by using the variational method, Liu and Huang obtained an existence result of sign-changing solutions as well as positive and negative solutions for a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity.

In this paper, we will study the existence of nontrivial solutions of problem (1). Our main results are the following theorems.

Theorem 1. Assume that 𝐹 is even in 𝑢 and the following conditions hold: (𝐹₁)𝐹(𝑥,𝑢)𝑢²⟶+∞𝑎𝑠|𝑢|⟶∞ (9) uniformly in 𝑥∈Ω;(𝐹₂)there exist two constants 2<𝜃<2𝑁/(𝑁−4)=2^∗∗ and 𝑘₁>0 such that 𝑓(𝑥,𝑢)≤𝑘₁1+|𝑢|^𝜃−1 (10) uniformly for all 𝑥∈Ω;(𝐹₃)there exists a constant 𝛾>𝑁(𝜃−2)/4 such that liminf_|𝑢|→∞𝑓(𝑥,𝑢)𝑢−2𝐹(𝑥,𝑢)|𝑢|^𝛾>0, (11) uniformly for all 𝑥∈Ω.Then problem (1) has infinitely many nontrivial solutions.

Theorem 2. Assume that 𝐹 satisfies (𝐹₁) and (𝐹₄)there exist three positive constants 𝐿, 𝑚₁, and 𝑚₂ such that(𝑗₁)𝑓(𝑥,𝑢)𝑢−2𝐹(𝑥,𝑢)≥𝑚₁|𝑢|², if |𝑢|≥𝐿;(𝑗₂)|𝑓(𝑥,𝑢)|^𝜎/|𝑢|^𝜎≤𝑚₂(𝑓(𝑥,𝑢)𝑢−2𝐹(𝑥,𝑢)), if |𝑢|≥𝐿, where 𝜎>(𝑁−2)/2.If 𝐹 is even in 𝑢, problem (1) has infinitely many nontrivial solutions.

Remark 3. For Schrödinger equation, the corresponding condition (𝐹₄) is due to Ding and Luan [11]. The condition (𝐹₄) is weaker than the usual Ambrosetti-Rabinowitz-type condition (see [11, 12]).

Theorem 4. Assume that 𝐹 satisfies (𝐹₁), (𝐹₂), (𝐹₃), and (𝐹₅)𝐹(𝑥,𝑢)𝑢²⟶0𝑎𝑠𝑢⟶0 (12) uniformly in 𝑥∈Ω.If 0 is an eigenvalue of Δ²+𝑐Δ+𝑎 (with Navier boundary condition), assume also the condition that (𝐹₆)there exists 𝛿>0 such that(i)𝐹(𝑥,𝑢)≥0,  for all |𝑢|≤𝛿,𝑥∈Ω; or(ii)𝐹(𝑥,𝑢)≤0,  for all |𝑢|≤𝛿,𝑥∈Ω. Then problem (1) has at least one nontrivial solution.

Theorem 5. Suppose that 𝐹 satisfies (𝐹₁), (𝐹₄), and (𝐹₅). If 0 is an eigenvalue of Δ²+𝑐Δ+𝑎 (with Navier boundary condition), assuming also (𝐹₆), then problem (1) has at least one nontrivial solution.

Here, we have 𝜑(𝑢)=12_Ω|Δ𝑢|²−𝑐|∇𝑢|²+𝑎(𝑥)𝑢²𝑑𝑥−_Ω𝐹()=1𝑥,𝑢𝑑𝑥2𝑢⁺²−‖𝑢⁻‖²−_Ω𝐹(𝑥,𝑢)𝑑𝑥, (13) where 𝑢⁻∈𝐸⁻,𝑢⁺∈𝐸⁺, and 𝐸⁻(𝐸⁺) is the space spanned by the eigenvectors corresponding to negative (positive) eigenvalues of Δ²+𝑐Δ+𝑎. It is easy to know that 𝜑∈𝐶¹(𝐸,ℝ) under the conditions of our theorems.

It is well known that 𝐸 is continuously embedded in 𝐿^𝜆(Ω) for every 𝜆∈[1,2𝑁/(𝑁−4)]. If 1≤𝜆<2𝑁/(𝑁−4), the embedding is compact. It follows from (𝐹₂), (𝐹₃), and (𝐹₄) that 𝜃<2𝑁,𝑁−4𝜃𝑁−4𝛾<𝑁−42𝑁,𝑁−42𝜎<𝜎−12𝑁.𝑁−4 (14) Hence, there is a positive constant 𝐾 such that ‖𝑢‖_𝐿^𝜆≤𝐾‖𝑢‖,∀𝑢∈𝐸 (15) for 𝜆=1,2,𝜃,2𝜎/(𝜎−1), 2𝑁/(𝑁−4)=2^∗∗, where ‖⋅‖_𝐿^𝜆 denotes the norm of 𝐿^𝜆(Ω).

2. Proof of Main Results

In this paper, we will use the Fountain Theorem of Bartsch ([13, Theorem 2.5], [14, Theorem 3.6]) to prove our Theorems 1 and 2. And, we will prove Theorems 4 and 5 by using a version of Local Linking theorem [12, Theorem 2.2] which extends theorems given by Li and Willem [15], Li and Szulkin [16].

In [13, 14], Bartsch established the Fountain Theorem under the (PS)_𝑐 condition. Since the Deformation Theorem is still valid under the Cerami condition, the Fountain Theorem is true under the Cerami condition. So, we have the following Fountain Theorem.

Let 𝑋 be a reflexive and separable Banach space. It is well known that there exist {𝑣_𝑛}_𝑛∈ℕ⊂𝑋,{𝜓_𝑛}_𝑛∈ℕ⊂𝑋^∗ such that(1)⟨𝜓_𝑛,𝑣_𝑚⟩=𝛿_𝑛,𝑚 where 𝛿_𝑛,𝑚=1 for 𝑛=𝑚 and 𝛿_𝑛,𝑚=0 for 𝑛≠𝑚.(2)span{𝑣_𝑛𝑛∈ℕ}=𝑋, span^𝜔∗{𝜓_𝑛∣𝑛∈ℕ}=𝑋^∗.

Let 𝑋_𝑗=ℝ𝑣_𝑗; then 𝑋=⨁_𝑗≥1𝑋_𝑗. We define 𝑌_𝑘=^𝑘_𝑗=1𝑋_𝑗,𝑍_𝑘=^∞_𝑗=𝑘𝑋_𝑗. (16)

Theorem A (Fountain theorem). Assume that 𝜑∈𝐶¹(𝑋,ℝ) satisfies the Cerami condition (𝐶), 𝜑(−𝑢)=𝜑(𝑢). If for almost every 𝑘∈ℕ, there exist 𝜌_𝑘>𝑟_𝑘>0 such that (𝐴₁)𝑎_𝑘∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝜑(𝑢)≤0, (17)(𝐴₂)𝑏_𝑘∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝑟𝑘}𝜑(𝑢)⟶∞,𝑘⟶∞, (18) then 𝜑 has an unbounded sequence of critical values.

For the reader's convenience, we state the following Local Linking theorem [12, Theorem 2.2]. Let 𝑋 be a real Banach space with 𝑋=𝑋¹⊕𝑋² and 𝑋^𝑗₀⊂𝑋^𝑗₁⊂𝑋^𝑗₂⊂⋯⊂𝑋^𝑗 such that 𝑋^𝑗=⋃_𝑛∈ℕ𝑋^𝑗_𝑛,𝑗=1,2. For every multi-index 𝛼=(𝛼₁,𝛼₂)∈ℕ², let 𝑋_𝛼=𝑋¹_𝛼1⊕𝑋²_𝛼2. We define that 𝛼≤𝛽⇔𝛼₁≤𝛽₁,𝛼₂≤𝛽₂. A sequence {𝛼_𝑛}⊂ℕ² is admissible if for every 𝛼∈ℕ² there is 𝑚∈ℕ such that 𝑛≥𝑚⇒𝛼_𝑛≥𝛼. We say that 𝜑∈𝐶¹(𝑋,ℝ) satisfies the (𝐶^∗) condition if every sequence {𝑢_𝛼𝑛} such that {𝛼_𝑛} is admissible and satisfies 𝑢_𝛼𝑛∈𝑋_𝛼𝑛,sup_𝑛𝜑𝑢_𝛼𝑛𝑢<∞,1+_𝛼𝑛𝜑^′_𝛼𝑛𝑢_𝛼𝑛⟶0 (19) contains a subsequence which converges to a critical point of 𝜑, where 𝜑_𝛼=𝜑|_𝑋𝛼.

Theorem B (see [12, Theorem 2.2]). Suppose that 𝜑∈𝐶¹(𝑋,ℝ) satisfies the following assumptions: (𝑖₁)𝑋≠{0} and 𝜑 has a local linking at 0; that is, for some 𝑟>0, 𝜑(𝑢)≥0,∀𝑢∈𝑋¹‖𝑢‖𝜑(𝑢)𝑤𝑖𝑡ℎ≤𝑟,≤0,∀𝑢∈𝑋²𝑤𝑖𝑡ℎ‖𝑢‖≤𝑟; (20)(𝑖₂)𝜑 satisfies (𝐶^∗) condition;(𝑖₃)𝜑 maps bounded sets into bounded sets;(𝑖₄)for every 𝑚∈ℕ,𝜑(𝑢)→−∞ as |𝑢|→∞, on 𝑢∈𝑋¹_𝑚⊕𝑋². Then 𝜑 has at least one nonzero critical point.

Now, we can give the proof of our theorems.

Proof of Theorem 1. At first, we claim that 𝜑 satisfies the Cerami condition (𝐶). Consider a sequence {𝑢_𝑛} such that 𝜑(𝑢_𝑛) is bounded and ‖𝜑^′(𝑢_𝑛)‖(1+‖𝑢_𝑛‖)→0 as 𝑛→∞. Then there exists a constant 𝑀₁>0 such that 𝜑𝑢_𝑛≤𝑀₁,1+𝑢_𝑛𝜑^′_𝑛𝑢_𝑛≤𝑀₁. (21) By a standard argument, we only need to prove that {𝑢_𝑛} is a bounded sequence in 𝐸. Otherwise, going if necessary to a subsequence, we can assume that ‖𝑢_𝑛‖→∞ as 𝑛→∞. From (𝐹₃), there exist two constants 𝑘₂,𝑘₃>0 such that 𝑓(𝑥,𝑢)𝑢−2𝐹(𝑥,𝑢)≥𝑘₂|𝑢|^𝛾−𝑘₃,∀(𝑥,𝑢)∈Ω×ℝ. (22) So, by (21) and (22), we have 3𝑀₁≥2𝜑𝑢_𝑛−𝜑^′𝑢_𝑛,𝑢_𝑛=_Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛−2𝐹𝑥,𝑢_𝑛𝑑𝑥≥𝑘_2Ω𝑢_𝑛^𝛾𝑑𝑥−𝑘₃|Ω| (23) which implies that _Ω𝑢_𝑛^𝛾𝑑𝑥<𝑘₄ (24) for all 𝑛∈ℕ and some positive constant 𝑘₄.
Since 𝑁𝛾>4(),𝑁𝜃−24(𝜃−2)<2𝑁()<𝑁+4𝜃−12𝑁.𝑁−4 (25) On the one hand, we consider the case 𝑁4(𝜃−2)<𝛾<2𝑁().𝑁+4𝜃−1 (26)
Putting 2()()𝛾𝛼=𝜃−1𝑁−𝑁+4()𝛾,2𝑁−𝑁−4 (27) one has 0<𝛼<1. Let 𝛾𝑝=𝜃−1−𝛼>1,𝑢_𝑛=𝑢⁺_𝑛+𝑢⁻_𝑛+𝑢⁰_𝑛∈𝐸⁺⊕𝐸⁻⊕𝐸⁰, (28) where 𝐸⁰=ker(Δ²+𝑐Δ+𝑎). We can obtain from Hölder's inequality, (15), and (24) that _Ω𝑢_𝑛^𝜃−1𝑢⁺_𝑛=𝑑𝑥_Ω𝑢_𝑛^{𝜃−1−𝛼}𝑢_𝑛^𝛼𝑢⁺_𝑛𝑑𝑥=_Ω𝑢_𝑛^𝛾/𝑝𝑢_𝑛^𝛼𝑢⁺_𝑛≤𝑑𝑥_Ω𝑢_𝑛^{𝛾/𝑝𝑝}𝑑𝑥^1/𝑝_Ω𝑢_𝑛^𝛼𝑢⁺_𝑛^𝑞𝑑𝑥^1/𝑞≤_Ω𝑢_𝑛^𝛾𝑑𝑥^1/𝑝_Ω𝑢_𝑛^{𝑞𝛼2∗∗/(𝑞𝛼)}𝑑𝑥^𝛼/2∗∗×_Ω𝑢⁺_𝑛^{𝑞2∗∗/𝑞}𝑑𝑥^1/2∗∗≤𝑘₄^1/𝑝𝑢_𝑛^𝛼_𝐿2^∗∗𝑢⁺_𝑛𝐿2^∗∗≤𝑘₄^1/𝑝𝐾^𝛼+1𝑢_𝑛^𝛼𝑢⁺_𝑛 (29) for all 𝑛, where 𝑞=𝑝/(𝑝−1)=2^∗∗/(𝛼+1).
By (𝐹₂) and (29), one has 𝜑^′𝑢_𝑛,𝑢⁺_𝑛=𝑢⁺_𝑛²−_Ω𝑓𝑥,𝑢_𝑛𝑢⁺_𝑛≥𝑢𝑑𝑥_+𝑛2−_Ω𝑓𝑥,𝑢_𝑛𝑢⁺_𝑛≥𝑢𝑑𝑥_+𝑛2−𝑘_1Ω𝑢_𝑛^𝜃−1𝑢⁺_𝑛+𝑢⁺_𝑛=𝑢𝑑𝑥_+𝑛2−𝑘_1Ω𝑢_𝑛^𝜃−1𝑢⁺_𝑛𝑑𝑥−𝑘_1Ω𝑢⁺_𝑛≥𝑢𝑑𝑥_+𝑛2−𝑘₁𝑘₄^1/𝑝𝐾^𝛼+1𝑢_𝑛^𝛼𝑢⁺_𝑛−𝑘₁𝐾𝑢⁺_𝑛 (30) for all 𝑛. Since 𝛼<1, we have 𝑢⁺_𝑛𝑢_𝑛⟶0,as𝑛⟶∞. (31)
On the other hand, if 𝛾 satisfies, 2𝑁()𝑁+4𝜃−1≤𝛾<2𝑁.𝑁−4 (32) then one sees 1≤𝛾/(𝛾−𝜃+1)≤2𝑁/(𝑁−4). So, we get ‖𝑢‖_𝐿^{𝛾/(𝛾−𝜃+1)}≤𝐾‖𝑢‖,∀𝑢∈𝐸. (33) It follows from (24) that ∫_Ω𝑢_𝑛^𝛾𝑑𝑥𝑢_𝑛⟶0,as𝑛⟶∞. (34) By (𝐹₂) and (33), we obtain 𝜑^′𝑢_𝑛,𝑢⁺_𝑛=𝑢⁺_𝑛²−_Ω𝑓𝑥,𝑢_𝑛𝑢⁺_𝑛≥𝑢𝑑𝑥_+𝑛2−_Ω𝑓𝑥,𝑢_𝑛𝑢⁺_𝑛≥𝑢𝑑𝑥_+𝑛2−𝑘_1Ω𝑢_𝑛^𝜃−1𝑢⁺_𝑛+𝑢⁺_𝑛=𝑢𝑑𝑥_+𝑛2−𝑘_1Ω𝑢_𝑛^𝜃−1𝑢⁺_𝑛𝑑𝑥−𝑘_1Ω𝑢⁺_𝑛≥𝑢𝑑𝑥_+𝑛2−𝑘_1Ω𝑢_𝑛^{𝜃−1𝛾/𝜃−1}𝑑𝑥^{(𝜃−1)/𝛾}×_Ω𝑢⁺_𝑛^{𝛾/(𝛾−𝜃+1)}𝑑𝑥^{(𝛾−𝜃+1)/𝛾}−𝑘₁𝐾𝑢⁺_𝑛=𝑢⁺_𝑛²−𝑘₁𝑢_𝑛𝐿^𝜃−1_𝛾⋅𝑢⁺_{𝑛𝐿𝛾/(𝛾−𝜃+1)}−𝑘₁𝐾𝑢⁺_𝑛≥𝑢⁺_𝑛²−𝑘₁𝐾𝑢_𝑛𝐿^𝜃−1_𝛾⋅𝑢⁺_𝑛−𝑘₁𝐾𝑢⁺_𝑛 (35) for all 𝑛. Note that 𝛾≥2𝑁(𝜃−1)/(𝑁+4) and 𝑁>4 imply that 𝛾>𝜃−1. So, it follows from (34) and the above expression that 𝑢⁺_𝑛𝑢_𝑛⟶0,as𝑛⟶∞. (36) Hence, we conclude from (31) and (36) that 𝑢⁺_𝑛𝑢_𝑛⟶0,as𝑛⟶∞. (37) Similarly for 𝑢⁻_𝑛, we get 𝑢⁻_𝑛𝑢_𝑛⟶0,as𝑛⟶∞. (38)
It follows from the equivalence of the norms on the finite dimensional space 𝐸⁰ that there exists 𝐾₁>0 such that ‖𝑢‖_∞≤𝐾₁‖𝑢‖,‖𝑢‖≤𝐾₁‖𝑢‖_𝐿¹,‖𝑢‖≤𝐾₁‖𝑢‖_𝐿²,∀𝑢∈𝐸⁰. (39) Putting 𝜏=2𝑁+(5−𝑁)𝛾+1, one has 1<2𝜏−𝛾<𝜏−12𝑁.𝑁−4 (40) It follows from (24), (39), and Hölder's inequality that 1𝐾²₁𝑢⁰_𝑛²≤_Ω𝑢⁰_𝑛²≤𝑑𝑥_Ω𝑢_𝑛²𝑑𝑥=_Ω𝑢_𝑛^𝛾/𝜏𝑢_𝑛^{2−𝛾/𝜏}≤𝑑𝑥_Ω𝑢_𝑛^𝛾𝑑𝑥^1/𝜏_Ω𝑢_𝑛^{(2𝜏−𝛾)/(𝜏−1)}𝑑𝑥^{(𝜏−1)/𝜏}≤𝑘₄^1/𝜏𝐾^{(2𝜏−𝛾)/𝜏}𝑢_𝑛^{(2𝜏−𝛾)/𝜏} (41) and consequently 𝑢⁰_𝑛𝑢_𝑛⟶0,as𝑛⟶∞. (42) Hence, by (37), (38), and (42), one sees 𝑢1=^𝑛𝑢_𝑛≤𝑢⁺_𝑛+𝑢⁰_𝑛+𝑢⁻_𝑛𝑢_𝑛⟶0 (43) as 𝑛→∞, which is a contradiction. So, we obtain that {𝑢_𝑛} is bounded in 𝐸. By a standard argument, we get that 𝜑 satisfies the condition (𝐶).
Let 𝐸=⊕_𝑗≥1𝑋_𝑗 with dim𝑋_𝑗<∞ for any 𝑗≥1. Set 𝑌_𝑘=^𝑘_𝑗=1𝑋_𝑗,𝑍_𝑘=^∞_𝑗=𝑘𝑋_𝑗. (44) Since dim(𝑌_𝑘)<∞, all the norms are equivalent. For 𝑢∈𝑌_𝑘, there exists a constant 𝐾₂>0 such that ‖𝑢‖≤𝐾₂‖𝑢‖_𝐿². (45) From condition (𝐹₁), there exists 𝑘₅>0 such that 𝐹(𝑥,𝑢)≥𝐾²₂|𝑢|²−𝑘₅,∀(𝑥,𝑢)∈Ω×ℝ. (46) For 𝑢∈𝑌_𝑘, it follows from (45) and (46) that 𝜑(𝑢)=12𝑢⁺²−‖𝑢⁻‖²−_Ω𝐹()≤1𝑥,𝑢𝑑𝑥2𝑢⁺²−‖𝑢⁻‖²−𝐾²₂‖𝑢‖²_𝐿2+𝑘₅|Ω|≤12𝑢⁺²−‖𝑢⁻‖²−𝑢⁺²+𝑢⁰²+𝑘₅|Ω|1=−2𝑢⁺²−12‖𝑢⁻‖²−𝑢⁰²+𝑘₅|Ω|1≤−2‖𝑢‖²+𝑘₅|Ω| (47) which implies that 𝜑(𝑢)⟶−∞as‖𝑢‖⟶∞,in𝑌_𝑘. (48) So, (𝐴₁) of Theorem A is satisfied for every 𝜌_𝑘>0 large enough.
Here, we obtain from (𝐹₂) that there exists a positive constant 𝑘₆ such that |𝐹(𝑥,𝑢)|≤𝑘₆1+|𝑢|^𝜃 (49) uniformly for all 𝑥∈Ω. Let us define 𝛽_𝑘=sup_{𝑢∈𝑍𝑘,‖𝑢‖=1}‖𝑢‖_𝐿^𝜃. (50) For 𝑘 large enough, one has 𝑍_𝑘⊂𝐸⁺. By (49), on 𝑍_𝑘, we have 𝜑(𝑢)=‖𝑢‖²2−_Ω𝐹()≥‖𝑢‖𝑥,𝑢𝑑𝑥₂2−𝑘₆‖𝑢‖^𝜃_𝐿𝜃−𝑘₆|Ω|≥‖𝑢‖²2−𝑘₆𝛽^𝜃_𝑘‖𝑢‖^𝜃−𝑘₆|Ω|. (51) Choosing 𝑟_𝑘=(𝑘₆𝜃𝛽^𝜃_𝑘)^1/(2−𝜃), we obtain, if 𝑢∈𝑍_𝑘 and ‖𝑢‖=𝑟_𝑘, 𝜑(𝑢)≥12−1𝜃𝑘₆𝜃𝛽^𝜃_𝑘^2/(2−𝜃)−𝑘₆|Ω|. (52) Since, by Lemma 3.8 of [14], 𝛽_𝑘→0 as 𝑘→∞, (𝐴₂) is proved. Hence, the proof is completed by using Fountain theorem.

Proof of Theorem 2. Firstly, we claim that 𝜑 satisfies the Cerami condition (𝐶). Consider a sequence {𝑢_𝑛} such that 𝜑(𝑢_𝑛) is bounded from above and ‖𝜑^′(𝑢_𝑛)‖(1+‖𝑢_𝑛‖)→0 as 𝑛→∞. By a standard argument, we only need to prove that {𝑢_𝑛} is a bounded sequence in 𝐸. For otherwise, we can assume that ‖𝑢_𝑛‖→∞ as 𝑛→∞.
From assumption (𝐹₄), there exist two positive constants 𝑚₃ and 𝑚₄, such that 𝑚₃≥2𝜑𝑢_𝑛−𝜑^′𝑢_𝑛,𝑢_𝑛=_Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛−2𝐹𝑥,𝑢_𝑛≥𝑑𝑥_Ω𝑚₁𝑢_𝑛²𝑑𝑥−𝑚₄|Ω|. (53) So, one has _Ω𝑢_𝑛²𝑑𝑥≤𝑚₅ (54) for all 𝑛 and some positive constant 𝑚₅.
Let 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖; then ‖𝑣_𝑛‖=1 and ‖𝑣_𝑛‖_𝐿𝑟≤𝐶_𝑟 for all 𝑟∈[1,2𝑁/(𝑁−4)). By (54), we have _Ω𝑣²_𝑛1𝑑𝑥=𝑢_𝑛^2Ω𝑢_𝑛²𝑚𝑑𝑥≤⁵𝑢_𝑛²⟶0 (55) as 𝑛→∞. So, for 𝑟∈(2,(2𝑁−4)/(𝑁−4)), it follows from Hölder's inequality and the above expression that _Ω𝑣_𝑛^𝑟𝑑𝑥≤_Ω𝑣_𝑛^2(𝑟−1)𝑑𝑥^1/2_Ω𝑣_𝑛²𝑑𝑥^1/2⟶0 (56) as 𝑛→∞. It follows from (39) that 𝜑^′𝑢_𝑛,𝑢⁺_𝑛−𝑢⁻_𝑛=𝑢⁺_𝑛−𝑢⁻_𝑛²−_Ω𝑓𝑥,𝑢_𝑛𝑢⁺_𝑛−𝑢⁻_𝑛=𝑢𝑑𝑥_𝑛2−𝑢⁰_𝑛²−_Ω𝑓𝑥,𝑢_𝑛𝑢⁺_𝑛−𝑢⁻_𝑛=𝑢𝑑𝑥_𝑛21−_Ω𝑓𝑥,𝑢_𝑛𝑢⁺_𝑛−𝑢⁻_𝑛𝑢_𝑛²𝑑𝑥−𝑢⁰_𝑛²≥𝑢_𝑛²1−_Ω𝑓𝑥,𝑢_𝑛𝑢⁺_𝑛−𝑢⁻_𝑛𝑢_𝑛²𝑑𝑥−𝐾²₁𝑢⁰_𝑛²_𝐿2. (57) Hence, we get 1−_Ω𝑓𝑥,𝑢_𝑛𝑢⁺_𝑛−𝑢⁻_𝑛𝑢_𝑛²𝑑𝑥=𝑜(1). (58) From (𝐹₄), (53), and (56), there exists a positive constant 𝑚₆ such that _Ω𝑓𝑥,𝑢_𝑛𝑢⁺_𝑛−𝑢⁻_𝑛𝑢_𝑛²𝑑𝑥≤2_Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛𝑣_𝑛²𝑑𝑥≤2_Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛^𝜎𝑑𝑥^1/𝜎×_Ω𝑣_𝑛^2𝜎′𝑑𝑥^1/𝜎′≤𝑚_6Ω𝑣_𝑛^2𝜎′𝑑𝑥^1/𝜎′⟶0 (59) as 𝑛→∞; therefore, 1=𝑜(1), which is a contradiction. Hence, {𝑢_𝑛} is bounded.
In a way similar to the proof of Theorem 1, we can obtain that 𝜑 satisfies (𝐴₁) of Theorem A.
It follows from (𝐹₁) that there is 𝐿₁>0 such that 𝐹(𝑥,𝑢)≥|𝑢|²,∀|𝑢|≥𝐿₁. (60) Then, by (𝐹₄) and (60), for |𝑢|≥𝐿₂=max{𝐿,𝐿₁},𝑥∈Ω, one has 𝑓(𝑥,𝑢)^𝜎≤𝑚₂𝑓(𝑥,𝑢)𝑢−2𝐹(𝑥,𝑢)|𝑢|^𝜎≤𝑚₂𝑓(𝑥,𝑢)|𝑢|^𝜎+1 (61) which implies that 𝑓(𝑥,𝑢)≤𝑚₂^1/(𝜎−1)|𝑢|^{(𝜎+1)/(𝜎−1)} (62) for |𝑢|≥𝐿₂.
Therefore, there exist two positive constants 𝑚₇ and 𝑚₈ such that |𝐹(𝑥,𝑢)|≤𝑚₇+𝑚₈|𝑢|^{2𝜎/(𝜎−1)},∀(𝑥,𝑢)∈Ω×ℝ, (63) where 2𝜎/(𝜎−1)<2𝑁/(𝑁−4). As the proof of Theorem 1, we can get (𝐴₂). Therefore, Theorem 2 holds.

Proof of Theorem 4. The proof of this theorem is divided in several steps.
Step  1. We claim that 𝜑 has a local linking at zero with respect to (𝑋¹,𝑋²).
By (𝐹₅), for any 𝜀>0, there exists 𝛿₁>0 such that |𝐹(𝑥,𝑢)|≤𝜀𝑢²,∀|𝑢|≤𝛿₁. (64) We obtain from the above expression and (49) that |𝐹(𝑥,𝑢)|≤𝜀𝑢²+𝑀₂|𝑢|^𝜃,∀(𝑥,𝑢)∈Ω×ℝ, (65) where 𝑀₂=𝑘₆+𝑘₆𝛿₁^−𝜃. Hence, we can get from (15) and (65) that _Ω𝐹(𝑥,𝑢)𝑑𝑥≤_Ω𝜀𝑢²𝑑𝑥+𝑀_2Ω|𝑢|^𝜃‖𝑢‖𝑑𝑥=𝜀²_𝐿2+𝑀₂‖𝑢‖^𝜃_𝐿𝜃≤𝐾²𝜀‖𝑢‖²+𝐾^𝜃𝑀₂‖𝑢‖^𝜃 (66) for all 𝑢∈𝐸.
Here, we consider only the case where 0 is an eigenvalue of Δ²+𝑐Δ+𝑎 and case (ii) of (𝐹₆) holds. The case (i) is similar.
Let 𝑋=𝐸,𝑋¹=𝐸⁺⊕𝐸⁰, and 𝑋²=𝐸⁻, where 𝐸⁰=ker(Δ²+𝑐Δ+𝑎). Choose a Hilbertian basis {𝑒_𝑛}_𝑛≥0 for 𝑋¹ and define 𝑋¹_𝑛=span𝑒₀,…,𝑒_𝑛𝑋,𝑛∈ℕ,_2𝑛=𝑋²𝑋,𝑛∈ℕ,₁=_𝑛𝑋¹_𝑛. (67) Now, by (66), for each 𝑢∈𝑋²=𝐸⁻, one has 𝜑(𝑢)1=−2‖𝑢‖²−_Ω𝐹()1𝑥,𝑢𝑑𝑥≤−2‖𝑢‖²+𝐾²𝜀‖𝑢‖²+𝐾^𝜃𝑀₂‖𝑢‖^𝜃. (68) Letting 𝜀=1/(8𝐾²) and by 𝜃>2, we have 𝜑(𝑢)≤0,∀𝑢∈𝑋²with‖𝑢‖≤𝛿₂ (69) for 𝛿₂>0 small enough.
Let 𝑢=𝑢⁰+𝑢⁺∈𝐸⁰⊕𝐸⁺=𝑋¹ be such that ‖𝑢‖≤𝛿₃∶=𝛿/(2𝐾₁). Put Ω₁=𝑥∈Ω∣𝑢⁺(𝑥≤𝛿)2,Ω₂=Ω⧵Ω₁. (70) Then, for all ‖𝑢‖≤𝛿₃ and 𝑥∈Ω, by (39), one sees 𝑢⁰(𝑥)≤𝑢⁰_∞≤𝐾₁𝑢⁰≤𝐾₁‖𝑢‖≤𝛿2. (71) On one hand, from above expression, for any 𝑥∈Ω₁, we have |𝑢(𝑥)|≤𝑢⁰(𝑥)+𝑢⁺(𝑥)≤𝑢⁰_∞+𝛿2≤𝛿. (72) Hence, by condition (ii) of (𝐹₆), we get _Ω1𝐹(𝑥,𝑢)𝑑𝑥≤0. (73) On the other hand, for any 𝑥∈Ω₂, one has |𝑢(𝑥)|≤𝑢⁰(𝑥)+𝑢⁺(𝑥≤𝛿)2+𝑢⁺(𝑥)≤2𝑢⁺(𝑥). (74) Hence, for all 𝑥∈Ω₂ and 𝑢∈𝑋¹ with ‖𝑢‖≤𝛿₃, we can obtain from (65) that 𝐹(𝑥,𝑢)≤𝜀𝑢²+𝑀₂|𝑢|^𝜃≤4𝜀𝑢⁺(𝑥)²+2^𝜃𝑀₂𝑢⁺(𝑥)^𝜃, (75) which implies that _Ω2𝐹(𝑥,𝑢)𝑑𝑥≤4𝜀_Ω2𝑢⁺(𝑥)²𝑑𝑥+2^𝜃𝑀_2Ω2𝑢⁺(𝑥)^𝜃𝑢𝑑𝑥=4𝜀⁺²_𝐿2+2^𝜃𝑀₂𝑢^+𝜃_𝐿𝜃≤4𝐾²𝜀𝑢⁺²+(2𝐾)^𝜃𝑀₂𝑢^+𝜃. (76) Letting 𝜀=1/(16𝐾²) in above expression, then for all 𝑥∈Ω₂ and 𝑢∈𝑋¹ with ‖𝑢‖≤𝛿₃, we have 𝜑(𝑢)=12𝑢⁺²−_Ω2𝐹(𝑥,𝑢)𝑑𝑥−_Ω1𝐹()≥1𝑥,𝑢𝑑𝑥2𝑢⁺²−4𝐾²𝜀𝑢⁺²−(2𝐾)^𝜃𝑀₂𝑢^+𝜃≥14𝑢⁺²−(2𝐾)^𝜃𝑀₂𝑢^+𝜃, (77) which implies that 𝜑(𝑢)≥0,∀𝑢∈𝑋¹with‖𝑢‖≤𝛿₄ (78) for 𝛿₄>0 small enough. Hence, 𝜑 has a local linking at zero with respect to (𝑋¹,𝑋²) for 𝛿₅=min{𝛿₂,𝛿₄} small enough.
Step  2. In a way similar to the proof of Theorem 1, we can get that 𝜑 satisfies the (𝐶^∗) condition.
Step  3. Now, we claim that for each 𝑚∈ℕ, one has 𝜑(𝑢)⟶−∞,as‖𝑢‖⟶∞,in𝑋¹_𝑚⊕𝑋². (79) Since dim(𝐸⁰)<∞ and dim(𝑋¹_𝑚)<∞, all the norms are equivalent. For 𝑢∈𝑋¹_𝑚⊕𝑋², there exists a constant 𝑘₇>0 such that ‖𝑢‖≤𝑘₇‖𝑢‖_𝐿². (80) From condition (𝐹₁), there exists a constant 𝑘₈>0 such that 𝐹(𝑥,𝑢)≥𝑘²₇|𝑢|²−𝑘₈,∀(𝑥,𝑢)∈Ω×ℝ. (81) For 𝑢∈𝑋¹_𝑚⊕𝑋², it follows from (80) and (81) that 𝜑(𝑢)=12𝑢⁺²−‖𝑢⁻‖²−_Ω𝐹()≤1𝑥,𝑢𝑑𝑥2𝑢⁺²−‖𝑢⁻‖²−𝑘²₇‖𝑢‖²_𝐿2+𝑘₈|Ω|≤12𝑢⁺²−‖𝑢⁻‖²−𝑘²₇𝑢⁺²_𝐿2+𝑢⁰²_𝐿2+𝑘₈|Ω|≤12𝑢⁺²−‖𝑢⁻‖²−𝑢⁺²+𝑢⁰²+𝑘₈|Ω|1=−2𝑢⁺²−12‖𝑢⁻‖²−𝑢⁰²+𝑘₈|Ω|1≤−2‖𝑢‖²+𝑘₈|Ω| (82) which implies that 𝜑(𝑢)⟶−∞as‖𝑢‖⟶∞,in𝑋¹_𝑚⊕𝑋². (83) Hence, all the assumptions of Theorem B are verified. Then, the proof of Theorem 4 is completed.

Proof of Theorem 5. In a way similar to the proof of Theorems 2 and 4, we can obtain that 𝜑 satisfies (𝑖₁), (𝑖₂), (𝑖₃), and (𝑖₄) of Theorem B. Therefore, Theorem 5 holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for the valuable suggestions. This paper is supported by the National Natural Science Foundation of China (no. 11071198) and the Fundamental Research Funds for the Central Universities (no. XDJK2010C055).