Sign-Changing Solutions for a Fourth-Order Elliptic Equation with Hardy Singular Terms

Abstract

The existence and multiplicity of sign-changing solutions for a class of fourth elliptic equations with Hardy singular terms are established by using the minimax methods.

1. Introduction

Consider the following Navier boundary value problem: △²𝑢(𝑥)−𝑁²(𝑁−4)²𝑢16|𝑥|⁴=𝑓(𝑥,𝑢),inΩ,𝑢=△𝑢=0on𝜕Ω, (1) where Ω is a bounded smooth domain in ℝ^𝑁(𝑁≥5), 0∈Ω.

The conditions imposed on 𝑓(𝑥,𝑡) are as follows:(𝐻₁) there exists 𝐶>0 such that 𝑓(𝑥,𝑡)≤𝐶1+|𝑡|^𝑝,∀𝑡∈ℝ,∀𝑥∈Ω, (2)where 1<𝑝<(𝑁+4)/(𝑁−4);(𝐻₂)𝑓∈𝐶(Ω×ℝ,ℝ),𝑓(𝑥,𝑡)  𝑡≥0 for all 𝑥∈Ω, 𝑡∈ℝ;(𝐻₃)lim_|𝑡|→0𝑓(𝑥,𝑡)/𝑡=𝑓₀, lim_|𝑡|→∞𝑓(𝑥,𝑡)/𝑡=𝑙 uniformly for 𝑥∈Ω, where 𝑓₀ and 𝑙 are constants;(𝐻₄)lim_|𝑡|→∞[𝑓(𝑥,𝑡)𝑡−2𝐹(𝑥,𝑡)]=−∞ uniformly for 𝑥∈Ω, where 𝐹(𝑥,𝑡)=∫^𝑡₀𝑓(𝑥,𝑠)𝑑𝑠;(𝐻₅) there exist 𝜇>2 and 𝑅>0 such that 0<𝜇𝐹(𝑥,𝑡)≤𝑓(𝑥,𝑡)𝑡,𝑥∈Ω,|𝑡|≥𝑅; (3)(𝐻₆)𝑓(𝑥,𝑡) is odd in 𝑡.

In recent years, this fourth-order semilinear elliptic problem: △²𝑢(𝑥)+𝑐△𝑢=𝑓(𝑥,𝑢),inΩ,𝑢=△𝑢=0on𝜕Ω, (4) can be considered as an analogue of a class of second-order problems which have been studied by many authors. In [1], there was a survey of results obtained in this direction. In [2], Micheletti and Pistoia showed that (4) admits at least two solutions by a variation of linking if 𝑓(𝑥,𝑢) is sublinear. And in [3], the authors proved that the problem (4) has at least three solutions by a variational reduction method and a degree argument. In [4], Zhang and Li showed that (4) admits at least two nontrivial solutions by Morse theory and local linking if 𝑓(𝑥,𝑢) is superlinear and subcritical on 𝑢.

To the authors’ knowledge, there seem few results about the sign-changing solutions on problem (1) with hardy singular terms. In this paper, motivated by [5–8], the existence and multiplicity of sign-changing solutions for problem (1) are obtained by introducing a compact embedding theorem and a maximum principle. Our results are new.

2. Preliminaries and Auxiliary Lemmas

We introduce the new working space 𝐸 which is obtained by the completion of 𝐶^∞₀(Ω) with respect to the norm (see [5]) ‖𝑢‖=_Ω|△𝑢|²−𝑁²(𝑁−4)²|𝑢|16|𝑥|²⁴𝑑𝑥^1/2 (5) associated with the inner product ⟨𝑢,𝑣⟩=_Ω𝑁△𝑢△𝑣−²(𝑁−4)²16𝑢𝑣|𝑥|⁴𝑑𝑥. (6)

Throughout this paper, we denoted by ‖⋅‖_𝑝 the 𝐿^𝑝(Ω) norm.

At first, we here give two important lemmas.

Lemma 1. 𝐸↪↪𝐿²(Ω) (see [5]).

Lemma 2 (see [6, Corollary 4.1]). Assume 𝑁≥5,  𝑉∈𝐿^∞(Ω), and 𝑉≥0. Let us suppose that the operator △²−(𝑉/|𝑥|⁴) is coercive on 𝐻²(Ω)∩𝐻¹₀(Ω). Let 𝑓∈𝐿²(Ω) such that 𝑓≥0. Let 𝑢∈𝐻²(Ω) be a solution of △²𝑢(𝑥)−𝑉|𝑥|⁴𝑢=𝑓,𝑖𝑛Ω,𝑢=△𝑢=0𝑜𝑛𝜕Ω. (7) Then 𝑢≥0 in Ω.

Now, we consider the following eigenvalue problem: △²𝑢(𝑥)−𝑁²(𝑁−4)²16|𝑥|⁴𝑢=𝜆𝑢,inΩ,𝑢=△𝑢=0on𝜕Ω. (8)(9)

The first eigenvalue of this problem is given by 𝜆₁=inf‖𝑢‖²∶𝑢∈𝐸,‖𝑢‖₂=1. (10)

By Lemma 1, 𝐸↪𝑊^1,𝑝(Ω)↪↪𝐿²(Ω) for 𝑝→2⁻. The minimizing sequence is compact in 𝐿²(Ω). By standard argument, we may assume that the first eigenfunction 𝜙₁ is positive in Ω (see [9, page 167]). The second eigenvalue is given by 𝜆₂=inf‖𝑢‖²∶𝑢∈𝐸,_Ω𝑢𝜙₁=0,‖𝑢‖₂=1 (11) which possesses a sign-changing eigenfunction 𝜙₂. Similarly, we can characterize the 𝑛th eigenvalue 𝜆_𝑛 with a sign-changing eigenfunction. By standard elliptic theory, 𝜆_𝑛→∞ as 𝑛→∞.

It follows from (𝐻₁) that the functional 𝐼(𝑢)=12_Ω|△𝑢|²𝑁𝑑𝑥−²(𝑁−4)²×32_Ω𝑢²|𝑥|⁴−_Ω𝐹(𝑥,𝑢)𝑑𝑥 (12) is of 𝐶¹ on the space 𝐸. Under the condition (𝐻₁), the critical points of 𝐼 are solutions of problem (1).

If 𝑙 in the above condition (𝐻₃) is an eigenvalue of (△²−(𝑁²(𝑁−4)²/16)(1/|𝑥|⁴),𝐸), then the problem (1) is called resonance at infinity. Otherwise, we call it nonresonance.

For looking for sign-changing solutions of problem (1), we recall a very useful result.

Proposition 3 (see [10, Theorem 3.2]). Let 𝑋 be a Hilbert space and 𝑓 be a 𝐶¹ functional defined on 𝑋. Assume that 𝑓 satisfies the (PS) condition on 𝑋 and 𝑓^′(𝑢) has the expression 𝑓^′(𝑢)=𝑢−𝐴𝑢 for 𝑢∈𝑋. Assume that 𝐷₁ and 𝐷₂ are open convex subset of 𝑋 with the properties that 𝐷₁∩𝐷₂≠∅, 𝐴(𝜕𝐷₁)⊂𝐷₁, and 𝐴(𝜕𝐷₂)⊂𝐷₂. If there exists a path ℎ∶[0,1]→𝑋 such that ℎ(0)∈𝐷₁⧵𝐷₂,ℎ(1)∈𝐷₂⧵𝐷₁,inf_{𝑢∈𝐷1∩𝐷2}𝑓(𝑢)>sup_[]𝑡∈0,1𝑓(ℎ(𝑡)), (13) then f has at least four critical points, one in 𝐷₁∩𝐷₂, one in 𝐷₁⧵𝐷₂, one in 𝐷₂⧵𝐷₁, and one in 𝑋⧵(𝐷₁∪𝐷₂).

Remark 4. If 𝑓 satisfies the (𝐶)_𝑐 condition, then this proposition still holds (see [11]).

3. Main Results

Let us now state the main results.

Theorem 5. Assume conditions (𝐻₂) and (𝐻₃) hold. If 𝑓₀<𝜆₁ and 𝑙∈(𝜆_𝑘,𝜆_𝑘+1) for some 𝑘>2, then problem (1) has a positive solution, a negative solution, and a sign-changing solution.

Remark 6. This result is similar to [7, Theorem1.1]. As far as verifying the (PS) condition is concerned, our method is more simple than that in [12, 13].

Theorem 7. Assume conditions (𝐻₂)–(𝐻₄) hold. If 𝑓₀<𝜆₁ and 𝑙=𝜆_𝑘 for some 𝑘>2, then problem (1) has a positive solution, a negative solution, and a sign-changing solution.

Remark 8. When 𝑙=𝜆_𝑘(𝑘>2), the case is called resonance and not considered by [7]. This result is completely new.

Theorem 9. Assume conditions (𝐻₁), (𝐻₅), and (𝐻₆) hold. If 𝑓₀=0, then problem (1) has infinitely many sign-changing solutions.

Lemma 10. Under the assumptions of Theorem 5, if 𝜆_𝑘<𝑙<𝜆_𝑘+1, then 𝐼 satisfies the (PS) condition.

Proof. Let {𝑢_𝑛}⊂𝐸 be a sequence such that |𝐼(𝑢_𝑛)|≤𝑐,  <𝐼^′(𝑢_𝑛), and   𝜙>→0. Since𝐼^′𝑢_𝑛=,𝜙_Ω△𝑢_𝑛𝑁△𝜙−²(𝑁−4)²𝑢16^𝑛𝜙|𝑥|⁴−𝑑𝑥_Ω𝑓𝑥,𝑢_𝑛𝜙𝑑𝑥=𝑜𝜙 (14) for all 𝜙∈𝐸. If ‖𝑢_𝑛‖₂ is bounded, we can take 𝜙=𝑢_𝑛. By (𝐻₃), there exists a constant 𝑐>0 such that |𝑓(𝑥,𝑢_𝑛(𝑥))|≤𝑐|𝑢_𝑛(𝑥)|, a.e. 𝑥∈Ω. So 𝑢_𝑛 is bounded in 𝐸. If ‖𝑢_𝑛‖₂→+∞, as 𝑛→∞, set 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖₂, then ‖𝑣_𝑛‖₂=1. Taking 𝜙=𝑣_𝑛 in (14), it follows that ‖𝑣_𝑛‖ is bounded. Without loss of generality, we assume 𝑣_𝑛⇀𝑣 in 𝐸, and then 𝑣_𝑛→𝑣 in 𝐿²(Ω). Hence, 𝑣_𝑛→𝑣 a.e. in Ω and |𝑣_𝑛|≤𝑞(𝑥) (𝑞(𝑥)∈𝐿²(Ω)). Dividing both sides of (14) by ‖𝑢_𝑛‖₂, for all 𝜙∈𝐸, we get _Ω△𝑣_𝑛𝑁△𝜙−²(𝑁−4)²𝑣16^𝑛𝜙|𝑥|⁴−𝑑𝑥_Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛2𝜙𝜙𝑑𝑥=𝑜𝑢_𝑛2. (15)
Then for a.e. 𝑥∈Ω, we have 𝑓(𝑥,𝑢_𝑛)/‖𝑢_𝑛‖₂→𝑙𝑣 as 𝑛→∞. In fact, if 𝑣(𝑥)≠0, by (𝐻₃), we have 𝑢_𝑛(𝑥)=𝑣_𝑛(𝑥)𝑢_𝑛2𝑓⟶+∞,𝑥,𝑢_𝑛𝑢_𝑛2=𝑓𝑥,𝑢_𝑛𝑢_𝑛𝑣_𝑛⟶𝑙𝑣. (16)
If 𝑣(𝑥)=0, we have 𝑓𝑥,𝑢_𝑛𝑢_𝑛2≤𝑐𝑣_𝑛⟶0. (17)
Since |𝑓(𝑥,𝑢_𝑛)|/‖𝑢_𝑛‖₂≤𝑐|𝑣_𝑛|≤𝑐𝑞(𝑥), by (15) and the Lebesgue dominated convergence theorem, we arrive at _Ω𝑁△𝑣△𝜙𝑑𝑥−²(𝑁−4)²16𝑣𝜙|𝑥|⁴−_Ω𝑙𝑣𝜙𝑑𝑥=0,∀𝜙∈𝐸. (18)
It is easy to see that 𝑣≢0. In fact, if 𝑣≡0, then ‖𝑣‖₂=0 contradicts lim_𝑛→∞‖𝑣_𝑛‖₂=‖𝑣‖₂=1. Hence, 𝑙 is an eigenvalue of (△²−(𝑁²(𝑁−4)²/16)(1/|𝑥|⁴),𝐸). This contradicts our assumption. Thus {𝑢_𝑛} is bounded. By standard argument (see the proof of our Lemma 12 below), {𝑢_𝑛}→𝑢 in 𝐸. The lemma is proved.

Lemma 11. Under the assumptions of Theorem 7, if 𝑙=𝜆_𝑘, then the functional 𝐼 satisfies the (𝐶) condition which is stated in [11].

Proof. Suppose 𝐼 satisfies𝐼𝑢_𝑛⟶𝑐∈ℝ,1+𝑢_𝑛𝐼^′𝑢_𝑛⟶0as𝑛⟶∞. (19)
In view of (𝐻₃), it suffices to prove that 𝑢_𝑛 is bounded in 𝐸. Similar to the proof of Lemma 10, we have _Ω𝑁△𝑣△𝜙−²(𝑁−4)²16𝑣𝜙|𝑥|⁴−𝑑𝑥_Ω𝑙𝑣𝜙𝑑𝑥=0,∀𝜙∈𝐸. (20)
Therefore 𝑣≢0 is an eigenfunction of 𝜆_𝑘, and then |𝑢_𝑛(𝑥)|→∞ for a.e. 𝑥∈Ω. It follows from (𝐻₄) that lim_𝑛→+∞𝑓𝑥,𝑢_𝑛(𝑥)𝑢_𝑛(𝑥)−2𝐹𝑥,𝑢_𝑛(𝑥)=−∞ (21) holds uniformly in 𝑥∈Ω, which implies that _Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛−2𝐹𝑥,𝑢_𝑛𝑑𝑥⟶−∞as𝑛⟶∞. (22)
On the other hand, (19) implies that 2𝐼𝑢_𝑛−𝐼^′𝑢_𝑛,𝑢_𝑛⟶2𝑐as𝑛⟶∞. (23)
Thus _Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛−2𝐹𝑥,𝑢_𝑛𝑑𝑥⟶2𝑐as𝑛⟶∞, (24) which contradicts (22). Hence 𝑢_𝑛 is bounded.

Lemma 12. Assume (𝐻₁) and (𝐻₅) hold. Then 𝐼 satisfies the (PS) condition.

Proof. Assume that {𝑢_𝑛} is a (PS) sequence; ‖𝐼^′(𝑢_𝑛)‖→0 and {𝐼(𝑢_𝑛)} is bounded. A routine argument implies that {‖𝑢_𝑛‖} is bounded. By [6, Theorem A.2], we have 𝐸↪𝑊₀^1,𝑞(Ω), (25) where 1≤𝑞<2. For 𝑝 given in (𝐻₁),  𝑝<(𝑛+4)/(𝑛−4), and we may choose 𝑞 such that (𝑝+1)<𝑞𝑁/(𝑁−𝑞),   𝑞<2. By the Sobolev embedding theorem, we have 𝑊^1,𝑞(Ω)↪↪𝐿^𝑡(Ω),∀𝑡<𝑁𝑞.𝑁−𝑞 (26) We infer from (26) that {𝑢_𝑛} is compact in 𝐿^𝑝+1(Ω). By (𝐻₁), 𝑢_𝑛−𝑢_𝑚²=_Ω𝑓𝑥,𝑢_𝑛−𝑓𝑥,𝑢_𝑚𝑢_𝑛−𝑢_𝑚𝑑𝑥+𝑜(1)≤𝐶_Ω|𝑢_𝑛−𝑢_𝑚|^𝑝+1𝑑𝑥^1/(𝑝+1)+𝑜(1)⟶0. (27)
This completes the proof of this lemma.

For the aim of using Proposition 3 that proves our main results, we prove an important lemma below.

From previous Section 1, we know that 𝐼 is 𝐶¹ functional and its gradient at 𝑢 is given by 𝐼^′(𝑢)(𝑢)𝐴(𝑢)=△=𝑢−𝐴,𝐴∶𝐸⟶𝐸,₂−𝑁²(𝑁−4)²116|𝑥|⁴⁻¹𝑓(𝑥,𝑢). (28)

Then ⟨𝐴(𝑢),𝜙⟩=∫_Ω𝑓(𝑥,𝑢)𝜙𝑑𝑥 for all 𝜙∈𝐸. We consider the convex cones 𝑃={𝑢∈𝐻∶𝑢≥0} and −𝑃={𝑢∈𝐸∶𝑢≤0}; moreover, for 𝜖>0, assume 𝑃_𝜖={𝑢∈𝐸∶dist(𝑢,𝑃)<𝜖},−𝑃_𝜖={𝑢∈𝐸∶dist(𝑢,−𝑃)<𝜖}. (29)

Note that 𝑃_𝜖 and −𝑃_𝜖 are open convex subsets of 𝐸  and 𝐸⧵(𝑃_𝜖∪(−𝑃_𝜖)) contains only sign-changing functions.

Lemma 13. Assume (𝐻₂) and (𝐻₃) hold. Then, there exists 𝜖₀>0 such that for 0<𝜖≤𝜖₀ there holds 𝐴𝜕±𝑃_𝜖⊂±𝑃_𝜖. (30)

Moreover, if 𝑢∈±𝑃_𝜖 is a nontrivial solution of problem (1), then 𝑢 is positive (negative) in the sense that 𝑢>0(𝑢<0) in Ω.

Proof. Indeed, if 𝑢∈𝐸 and 𝑢⁺=max{𝑢,0}, 𝑢⁻=min{𝑢,0}, then dist(𝐴(𝑢),𝑃)≤inf_𝑤∈𝑃‖𝐴(𝑢)−𝑤‖=inf_𝑤∈𝑃𝐴(𝑢)⁺+𝐴(𝑢)⁻−𝑤≤‖𝐴(𝑢)⁻‖. (31)
For every 𝑠∈(2,2𝑁/(𝑁−4)), there exists 𝐶_𝑠>0 such that 𝑢^±_𝑠≤inf_{𝑤∈∓𝑃}‖𝑢−𝑤‖_𝑠≤𝐶_𝑠dist(𝑢,∓𝑃). (32)
Choose 𝜖^′>0 such that (𝑓₀+𝜖^′)<𝜆₁. Using (32), the Hölder inequality, the Poincaré inequality, and the Sobolev embedding theorem, we have dist(𝐴(𝑢),𝑃)‖𝐴(𝑢)⁻‖≤‖𝐴(𝑢)⁻‖²=_Ω𝑓(𝑥,𝑢)𝐴(𝑢)⁻≤𝑑𝑥_Ω𝑓(𝑥,𝑢⁻)𝐴(𝑢)⁻≤𝑑𝑥_Ω𝑓₀+𝜖^′|𝑢⁻|+𝐶^′_𝜖|𝑢⁻|^𝑝𝐴(𝑢)⁻≤𝑓𝑑𝑥₀+𝜖^′‖𝑢⁻‖₂‖𝐴(𝑢)⁻‖₂+𝐶^′_𝜖‖𝑢⁻‖^𝑝_𝑝+1‖𝐴(𝑢)⁻‖_𝑝+1≤𝑓₀+𝜖^′inf_𝑤∈𝑃‖𝑢−𝑤‖₂‖𝐴(𝑢)⁻‖₂+𝐶inf_𝑤∈𝑃‖𝑢−𝑤‖^𝑝_𝑝+1‖𝐴(𝑢)⁻‖_𝑝+1≤𝑓₀+𝜖^′𝜆₁dist(𝑢,𝑃)‖𝐴(𝑢)⁻‖+𝐶dist(𝑢,𝑃)^𝑝‖𝐴(𝑢)⁻‖, (33) where 𝐶^′_𝜖,𝐶>0 are constants. Hence dist(𝐴(𝑢),𝑃)≤𝛿+𝐶dist(𝑢,𝑃)^𝑝−1dist(𝑢,𝑃), (34) where 𝛿=(𝑓₀+𝜖^′)/𝜆₁<1. Take 𝜖₀ such that 𝛿₁=𝛿+𝐶𝜖₀^𝑝−1<1. Now if dist(𝑢,𝑃)<𝜖<𝜖₀, then we have dist(𝐴(𝑢),𝑃)≤𝛿₁dist(𝑢,𝑃). (35)
Thus for every 𝑢∈𝜕𝑃_𝜖, by (35) we have dist(𝐴(𝑢),𝑃)≤𝛿₁𝜖; (36) thus 𝐴(𝑢)∈𝑃_𝜖. Hence 𝐴(𝜕𝑃_𝜖)⊂𝑃_𝜖. In a similar way, 𝐴(𝜕(−𝑃_𝜖))⊂(−𝑃_𝜖). If 0<𝜖≤𝜖₀, and 𝑢∈𝑃_𝜖 (resp.,—𝑃_𝜖) is a nontrivial solution of problem (1), then 𝐼^′(𝑢)=0. By (35) we have dist(𝑢,𝑃)=0; that is, 𝑢∈𝑃 (resp., 𝑢∈−𝑃). By Lemma 2, we imply that 𝑢>0(𝑢<0) in Ω.

Lemma 14. Assume (𝐻₁),   (𝐻₂),     and   (𝐻₅) hold. Then, there exists 𝜖₀>0 such that for 0<𝜖≤𝜖₀ there holds 𝐴𝜕±𝑃_𝜖⊂±𝑃_𝜖. (37)

Proof. The proof is quite similar to that of Lemma 4.2 in [8]. We omit it here.

Lemma 15. Assume (𝐻₅) holds. Then 𝐼(𝑢)⟶−∞,∀𝑢∈𝐸_𝑘, (38) where the definition of 𝐸_𝑘 introduced in our proof of Theorem 9.

Proof. Because dim𝐸_𝑘<∞, then by (𝐻₅), 𝐼(𝑢)‖𝑢‖²≤12−_Ω𝐹(𝑥,𝑢)‖𝑢‖²𝑑𝑥⟶−∞ (39) as ‖𝑢‖→∞,𝑢∈𝐸_𝑘. This lemma follows immediately.

Lemma 16. Assume (𝐻₂) and (𝐻₃) hold. Let 0<𝜖≤𝜖₀, and then there exists 𝐶₀>−∞ such that inf_{𝑃𝜖∩(−𝑃𝜖)}𝐼(𝑢)=𝐶₀.

Proof. By the conditions (𝐻₂) and (𝐻₃), we know that, for any 𝜖^′>0, there exists 𝐶>0, such that 𝑓(𝑥,𝑡)≤𝑓₀+𝜖^′|𝑡|+𝐶|𝑡|^𝑝1<𝑝<𝑁+4.𝑁−4 (40)
Using (40) and the Sobolev embedding theorem, we have 𝐼(𝑢)=12‖𝑢‖²−_Ω𝐹()≥1𝑥,𝑢𝑑𝑥2‖𝑢‖²−12𝑓₀+𝜖^′_Ω𝑢²𝑑𝑥−𝐶‖𝑢‖^𝑝+1_𝑝+11≥−2𝑓₀+𝜖^′‖𝑢‖²₂−𝐶‖𝑢‖^𝑝+1_𝑝+1. (41)
By (32) we have ‖𝑢^±‖_𝑠≤𝐶_𝑠𝜖₀ for every 𝑢∈𝑃_𝜖∩(−𝑃_𝜖). So there exists 𝐶₀>−∞ such that inf_{𝑃𝜖∩−𝑃𝜖}𝐼(𝑢)=𝐶₀. (42)
Hence this lemma is proved.

4. Proof of the Main Results

Proof of Theorem 5 and Theorem 7. Motivated by the Proof of Theorem 4.2 in [10], we still define a path ℎ_𝑅∶[0,1]→𝐸 as ℎ_𝑅(𝑡)=𝑅𝜙₁cos𝜋𝑡+𝑅𝜙₂sin𝜋𝑡,0≤𝑡≤1. (43)
Obviously, ℎ_𝑅(0)∈𝑃_𝜖⧵(−𝑃_𝜖) and  ℎ_𝑅(1)∈(−𝑃_𝜖)⧵𝑃_𝜖. By the Fatou’s lemma, the condition (𝐻₃) with 𝑙>𝜆₂ and a direct computation shows that lim_𝑅→+∞sup_[]𝑡∈0,1𝐼ℎ_𝑅(𝑡)=−∞. (44)
So, it yields that there exists 𝑅₀ such that 𝐼(ℎ_𝑅0(𝑡))<𝐶₀−𝑐^∗(𝑐^∗>0). Hence we obtain inf_{𝑃𝜖∩−𝑃𝜖}𝐼(𝑢)>sup_[]𝑡∈0,1𝐼(ℎ(𝑡)). (45) By using Lemmas 10, 11, and 13, Proposition 3, and Lemma 16, we can find a critical point in 𝑃_𝜖⧵(−𝑃_𝜖) which is a positive solution, a critical point in (−𝑃_𝜖)⧵𝑃_𝜖 which is a negative solution, and a critical point in 𝐸⧵(𝑃_𝜖∪(−𝑃_𝜖)) which is a sign-changing solution.

Before beginning our proof of Theorem 9, we need the following important proposition.

Proposition 17 (see [9, Theorem 5.6]). Assume 𝐸 is a Hilbert space with inner product <, > and the corresponding norm ‖⋅‖, 𝐼∈𝐶¹(𝐸,𝑅) and 𝐼(𝑢)=(1/2)‖𝑢‖²−𝐺(𝑢), 𝑢∈𝐸, where 𝐺∈𝐶¹(𝐸,𝑅). 𝑃 denotes a positive closed convex cone of 𝐸. (𝐴)Assume that   𝐴(±𝐷₀)⊂±𝐷₀, where 𝐷₀∶={𝑢∈𝐸∶dist(𝑢,𝑃)<𝜇₀}, 𝜇₀>0, and 𝐴=𝐺^′.(𝐴^∗₁)Assume that, for any 𝑎,𝑏>0, there is a constant 𝐶>0 such that𝐺(𝑢)≤𝑎,‖𝑢‖_∗≤𝑏⟹‖𝑢‖≤𝐶, (46)where ‖⋅‖_∗ denotes another norm of 𝐸 such that ‖𝑢‖_∗≤𝐶‖𝑢‖ for all 𝑢∈𝐸.(𝐴^∗₂)Assume that lim_{𝑢∈𝑌,‖𝑢‖→∞}𝐼(𝑢)=−∞,sup_𝑌𝐼∶=𝛽. If the even functional 𝐼 satisfies (𝑃𝑆) condition at level 𝑐 for each 𝑐∈[𝛾,𝛽], then 𝒦𝛾−𝜖,𝛽+𝜖∩(𝐸⧵(−𝑃∪𝑃))≠∅ (47) for all 𝜖>0 small, where (sup_𝑌𝐼∶=𝛽  (𝑌𝑎𝑛𝑑𝑀 are two subspaces of 𝐸 with dim𝑌<∞, dim𝑌−codim𝑀≥1),  inf_𝑄^∗∗𝐼∶=𝛾, and 𝑄^∗∗∶=𝑄^∗(𝜌)∩𝐼^𝛽(𝑄^∗(𝜌)∶={𝑢∈𝑀∶(‖𝑢‖^𝑝_∗/‖𝑢‖²)+((‖𝑢‖‖𝑢‖_∗)/(‖𝑢‖+𝐷_∗‖𝑢‖_∗))=𝜌}, where 𝜌>0, 𝐷_∗>0,  and 𝑝>2 are fixed constants.

Now, we give an outline proof for our Theorem 9.

Proof of Theorem 9. Let 𝑁_𝑘 denote the eigenspace of 𝜆_𝑘. We fix 𝑘 and let 𝐸_𝑘∶=𝑁₁⊕⋯⊕𝑁_𝑘. Consider another norm ‖⋅‖_∗∶=‖⋅‖_𝑝+1 of 𝐸, 𝑝∈(1,((𝑁+4)/(𝑁−4))). Write 𝐸=𝐸_𝑘−1⊕𝐸^⟂_𝑘−1.
Let 𝑄^∗𝜌∶=𝑢∈𝐸^⟂_𝑘−1∶‖𝑢‖^𝑝+1_𝑝+1‖𝑢‖²+‖𝑢‖‖𝑢‖_𝑝+1‖𝑢‖+𝐷_∗‖𝑢‖_𝑝+1=𝜌, (48) where 𝜌,𝐷_∗ are fixed constants. By our assumptions, we may find a constant 𝐶>0 such that 𝐹()≤1𝑥,𝑡4𝜆₁𝑡²+𝐶|𝑡|^𝑝+1,∀𝑥∈Ω,𝑡∈𝑅, (49) where 1<𝑝<(𝑁+4)/(𝑁−4). For any 𝑎,𝑏>0, there is a constant 𝐶>0 such that 𝐼(𝑢)≤𝑎,‖𝑢‖_𝑝+1≤𝑏⟹‖𝑢‖≤𝐶. (50) By Lemma 15, lim_{𝑢∈𝑌,‖𝑢‖→∞}𝐼(𝑢)=−∞, (51) where 𝑌=𝐸_𝑘. Then (𝐴^∗₁) and (𝐴^∗₂) are satisfied. By Lemma 14, the condition (𝐴) holds.
Now, we define sup_𝑌𝐼∶=𝛽. (52) Let 𝑄^∗∗∶=𝑄^∗𝜌∩𝐼^𝛽,inf_𝑄∗∗𝐼∶=𝛾. (53) By Lemma 12, 𝐼 satisfies the (PS) condition. Thus, by Proposition 17 and the Proof of Theorem 5.7 in [9], we know that the functional 𝐼 posses a sequence sign-changing solution {𝑢_𝑘}.

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant no. 10671156) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant no. 1301038C).