AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 267052 10.1155/2014/267052 267052 Research Article Multiple Results to Some Biharmonic Problems http://orcid.org/0000-0002-9062-1892 Tang Xingdong Zhang Jihui Hao Xinan Mathematical Sciences Nanjing Normal University No. 1 Wenyuan Road Yadong New District, Nanjing China nnu.cn 2014 15 4 2014 2014 05 12 2013 30 03 2014 15 4 2014 2014 Copyright © 2014 Xingdong Tang and Jihui Zhang. 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.

We study a nonlinear elliptic problem defined in a bounded domain involving biharmonic operator together with an asymptotically linear term. We establish at least three nontrivial solutions using the topological degree theory and the critical groups.

1. Introduction

We consider the following biharmonic problem: (1) Δ 2 u = f ( x , u ) in Ω , u = Δ u = 0 , on Ω , where Ω N    ( N 5 ) is a smooth bounded domain and f : Ω × is of class C 1 with f ( x , 0 ) = 0 .

In the past decades, biharmonic operators have attracted much attention of many researchers and experts. While f ( x , u ) = b [ ( u + 1 ) + - 1 ] , the solutions of (1) characterized the travelling waves in a suspension bridge; see .

In 1998, Micheletti and Pistoia  considered the following biharmonic problem: (2) Δ 2 u + a 2 Δ u = b [ ( u + 1 ) + - 1 ] , in Ω , u = Δ u = 0 , on Ω , where a , b are constants and Ω n is a bounded smooth domain, and they established multiple results by using a minimax process.

Three years later, Zhang  considered a more general condition; that is, (3) Δ 2 u + c Δ u = f ( x , u ) , in Ω , u = Δ u = 0 , on Ω , where c and f satisfies the subcritical growth; at least one nontrivial solution was obtained.

Since then, a lot of papers dealing with biharmonic problems by the critical point theory sprung up, and so forth [4, 5].

At the same time, Leray-Schauder degree as a very wonderful tool was introduced to handle biharmonic problems; see . To our best knowledge, there are few papers considered (1) by combining the critical point theory (especially Morse theory) with Leray-Schauder degree.

Our argument was originally developed by Hofer  and Zhang . Following Hofer  and Zhang , there are some papers dealing with second-order elliptic problems, and so forth .

Zhang  first considered the following second-order elliptic problem: (4) - Δ u = g ( x , u ) , in Ω , B u = 0 , on Ω , where B denotes Neumann operator or Dirichlet operator. Sub- and sup-solutions methods with critical point theory were used to obtain at least two distinct solutions. Also under subcritical growth condition, Chang  proved that if p 0 is an isolated critical point of J , then, for all q , C q ( J ~ , p 0 ) = C q ( J , p 0 ) with integral coefficients, where J ~ , J denote the energy functional under C 0 ( Ω ¯ ) C 1 ( Ω ¯ ) and H 0 1 ( Ω ) , C q means q th critical group corresponding to (4), which inspires us to consider (1).

Bartsch et al.  considered (4) and obtained more results in this direction. Then, some other results in this direction were also obtained; see .

As far as we know, there are few papers concerned with the biharmonic problem (1) using this method; only Qian and Li  considered (5) Δ 2 u + c Δ u = f ( x , u ) , in Ω , u = Δ u = 0 , on Ω ; And they proved that if u 0 is an isolated critical point of J , then, for all q , C q ( J ~ , u 0 ) = C q ( J , u 0 ) with integral coefficients, where J ~ , J denote the energy functional on the space C 0 ( Ω ¯ ) C 1 ( Ω ¯ ) and H 0 1 ( Ω ) H 2 ( Ω ) , C q means q th critical group. In our paper, the results in  are improved, and some new results are obtained.

Let 0 < λ 1 < λ 2 λ 3 λ n denote the eigenvalues of ( - Δ , H 0 1 ( Ω ) ) (counting with their multiplicity) with corresponding eigenfunctions e 1 , e 2 , e 3 , , e n , . We may choose e 1 > 0 in Ω . Let μ k = λ k 2 , k = 1,2 , , n , , then μ 1 < μ 2 μ 3 μ n are eigenvalues of the following biharmonic problem  corresponding eigenfunctions e 1 , e 2 , e 3 , , e n , ; (6) Δ 2 u = μ u , in Ω , u = Δ u = 0 , on Ω .

In order to obtain nontrivial solutions, we now assume that the nonlinearity f satisfies the following conditions:

f C 1 ( Ω ¯ × , ) , f ( x , u ) u 0 for all ( x , u ) Ω ¯ × , and there exist constant numbers C > 0 and α with 1 < α < ( N + 4 ) / ( N - 4 ) , such that (7) | f u ( x , u ) | c ( 1 + | u | α - 1 ) ;

there exists i with μ 2 i < μ 2 i + 1 , such that f u ( x , 0 ) = μ 2 i , for all x Ω ¯ ;

limsup | t | f ( x , u ) / u < μ 1 uniformly for x Ω ¯ ;

there is some r > 0 small, such that (8) μ 2 i u 2 F ( x , u ) < μ 2 i + 1 u 2 , u , | u | r , a . e . x Ω ,

where F ( x , u ) = 0 u f ( x , s ) d s .

The main result of this paper is the following

Theorem 1.

Suppose f satisfies (f1)–(f4). Then (1) has at least three solutions.

2. Preliminaries

In this section, we first recall some lemmas and preliminaries.

Let C 0 ( Ω ¯ ) C k ( Ω ¯ ) denote the set of f : Ω ¯ which are k -times continuous differentiable in Ω ¯ and identically vanishing on Ω with the norm u k = i = 0 k u ( k ) 0 , where u 0 = max x Ω ¯ u ( x ) , P k = { u C 0 ( Ω ¯ ) C k ( Ω ¯ ) : u ( x ) 0 , x Ω } , k .

Lemma 2.

P 2 is a solid cone of C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) ; that is, P ° 2 .

It is well known that the positive cone P 1 is a solid cone of C 0 ( Ω ¯ ) C 1 ( Ω ¯ ) . Our proof depends on the fact above; what is more, the technique we used here is originated from [15, page 628].

Proof.

Since P 1 is a closed positive cone of ( C 0 ( Ω ¯ ) C 1 ( Ω ¯ ) , · 1 ) , by the definition of · k , k = 0,1 , 2 , , n , , for u C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) , u 1 u 2 , thus the embedding  i : ( C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) , · 2 ) ( C 0 ( Ω ¯ ) C 1 ( Ω ¯ ) , · 1 ) is continuous. i - 1 ( P 1 ) is closed in ( C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) , · 2 ) (in fact P 2 = i - 1 ( P 1 ) ). Obviously P ° 1 ( C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) , · 2 ) ; thus P 2 = i - 1 ( P 1 ) has nonempty interior. The proof is finished.

Remark 3.

Using the method above, it is not difficult to know that P k is a solid cone in ( C 0 ( Ω ¯ ) C k ( Ω ¯ ) , · k ) , k = 2,3 , 4 , , n , .

Remark 4.

For any u C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) , if u is an interior point of P 1 in ( C 0 ( Ω ¯ ) C 1 ( Ω ¯ ) , · 1 ) , then u is an interior point of P 2 in ( C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) , · 2 ) .

In what follows, we will use the Hilbert space V = H 0 1 ( Ω ) H 2 ( Ω ) , and the norm on V is given by u V = Ω | Δ u | 2 d x . It is well known that solutions of (1) are critical points of the functional (9) Ψ ( u ) = 1 2 Ω | Δ u | 2 d x - Ω F ( x , u ) d x , where F ( x , u ) = 0 u f ( x , s ) d s . Since f C 1 ( Ω ¯ × , ) , it is easy to know that Ψ C 2 ( V , ) , and (10) Ψ ( u ) , v = Ω [ Δ u Δ v - f ( x , u ) v ] d x , Ψ ′′ ( u ) v , h = Ω [ Δ v Δ h - f ( x , u ) v h ] d x .

Corresponding to the eigenvalues μ j s we have the splitting V = H - N H + where (11) H - = j = 1 2 i - 1 e j , N = span { e 2 i } , H + = j = 2 i + 1 + e j ¯ .

Consider the problem (12) Δ 2 u = h , in Ω , u = Δ u = 0 , on Ω .

For all r + , denote B ( 0 , r ) { u C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) : u 2 < r } , U ( 0 , r ) { u V : u V < r } , P 2 r = P 2 B ( 0 , r ) , P 2 r = P 2 B ( 0 , r ) .

Let K denote the solution operator of (12), and ( f u ) ( x ) f ( x , u ( x ) ) . Under condition (f1), it is easy to see that A K f : V V is of class C 1 . Since f : V V * is completely continuous , then A : V V is completely continuous.

Lemma 5 (see [<xref ref-type="bibr" rid="B3">16</xref>]).

Suppose h L q ( Ω ) , q 2 ; then the weak solution u = K ( h ) of (12) satisfies u W 4 , q C h L q ; what is more, we have that (13) K : L q ( Ω ) W 4 , q ( Ω ) W 0 1 , q ( Ω ) is continuous.

Remark 6.

Actually, for all h V * , there exists a unique weak solution u = K ( h ) V of (12). Since by Riesz representation theorem, for all h V * , there exists a unique Θ = Θ ( h ) such that h , v = ( Θ , v )    v V ; thus Θ = K ( h ) is the corresponding weak solution.

Consider the Cauchy problem in V , (14) d d t u ( t ) = - u ( t ) + K f u ( t ) , u ( 0 ) = u 0 .

Lemma 7 (see [<xref ref-type="bibr" rid="B6">13</xref>]).

Let H be a real Hilbert space, and let ψ C 2 ( H , ) satisfy the ( P S ) condition. Assume that (15) ψ ( v ) = v - A v , v H , where A is a compact mapping, and that p 0 is an isolated critical point of f . Then we have (16) ind ( ψ , p 0 ) = q = 0 ( - 1 ) q rank C q ( ψ , p 0 ) .

Let X be a retract of a real Banach space E , let U be a relatively open subset of X , and let A : U ¯ X X be a completely continuous operator. Suppose that A has no fixed points on X U and that the fixed point of A is bounded. The following lemma establishes the relationship of fixed point index and topological degree.

Lemma 8 (see [<xref ref-type="bibr" rid="B19">17</xref>]).

If any fixed point of in U is an interior point of X , then there exists an open subset O of E with O U such that O contains all fixed points of A in U and (17) deg ( I - A , O , 0 ) = i ( A , U , X ) .

Remark 9.

Let O be a bounded open subset of U , and let there be no zero points of I - A on O . Since C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) can be compactly embedded into V , it follows from the bootstrap argument and the definition of Leray-Schauder degree that (18) de g V ( I - A , O , 0 ) = de g C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) ( I - A , O , 0 ) .

In what follows, de g C 0 2 ( Ω ¯ ) is denoted simply by deg .

Remark 10 (see [<xref ref-type="bibr" rid="B9">18</xref>]).

Remark 9 implies that two topological degrees in both de g C 0 2 ( Ω ¯ ) and de g V are the same. Combining with Lemma 7, we can obtain the connection between the topological degree and the critical group: (19) deg ( I - A , O , 0 ) = q = 0 ( - 1 ) q rank C q ( J , p 0 ) .

Lemma 11.

Let u ( t , u 0 ) be the unique solution of (14) with the maximal interval [ 0 , η ( u 0 ) ) . We have the following conclusions.

If  u 0 C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) , then { u ( t , u 0 ) : 0 t < η ( u 0 ) } C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) , and u ( t , u 0 ) is continuous as a function of t from [ 0 , η ( u 0 ) ) to C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) .

If u 0 , u C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) , u = K f u , and u ( t , u 0 ) - u 2 0 as t η ( u 0 ) , then u ( t , u 0 ) - u 2 0 as t η ( u 0 ) .

If u 0 C 0 ( Ω ¯ ) C 2 , μ ( Ω ¯ ) for some μ ( 0,1 ) then { u ( t , u 0 ) : 0 t < η ( u 0 ) } C 0 ( Ω ¯ ) C 2 μ ( Ω ¯ ) and is bounded in the C 0 2 , μ norm.

Lemma 11 essentially comes from .

Proof.

We only need to construct the embedding chains like (5) and (6) of ; the rest can be proved similar to [19, Lemma 2].

Without loss of generality, α can be assumed to satisfy max { 8 / ( N - 4 ) , 1 } < α < ( N + 4 ) / ( N - 4 ) . We can choose δ > 0 , such that (20) α < δ + ( N + 4 ) ( 1 - δ ) N - 4 . Let q 0 = 2 N / ( N - 4 ) , and define q i by (21) 1 q i + 1 = α q i - 2 N , i = 0,1 , 3 , .

A direct computation shows that (22) q n ( 5 5 - 4 δ ) n q 0 . Hence there exists a number n 3 such that (23) q 0 < q 1 < < q n - 3 < N α 2 q n - 2 .

Let (24) q i = q i , i = 0,1 , 2 , , n - 3 , and choose q n - 2 and q n - 1 such that (25) q n - 3 < q n - 2 < N α 2 , q n - 1 = α N .

Let (26) p i = q i α , i = 0,1 , 2 , , n - 1 . Define (27) X 0 = L q 0 ( Ω ) , X i + 1 = W 4 , p i ( Ω ) W 0 1 , p i , Y i = L p i ( Ω ) , Z i = L q i ( Ω ) , i = 0,1 , , n - 1 . Then we have the following imbedding chains: (28) V X n X n - 1 X n - 2 X 1 X 0 C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) Z n - 1 Z n - 2 Z 1 Z 0 , Y n - 1 Y n - 2 Y 1 Y 0 . What is more, we have the chains of bounded and continuous operators (29) Z i f Y i K X i + 1 , i = 0,1 , 2 , , n - 1 .

Lemma 12.

Suppose that (f1) and (f3) hold. Then Ψ satisfies the ( P S ) condition.

The proof of this lemma is similar to the proof of [5, Lemma 2.1]. We omit it here.

Since A : V V is completely continuous, then by the above bootstrap iteration, A : C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) is completely continuous. For our application, sometimes we would consider the restriction Ψ ~ of Ψ on a smaller Banach space C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) . The functional may lose the (PS) condition. However following , the following two lemmas can be obtained.

Lemma 13 (see [<xref ref-type="bibr" rid="B4">20</xref>]).

Suppose that (f1) and (f3) hold. Then Ψ ~ possesses the following properties.

Ψ ~ ( K ) is a closed subset.

For each pair a < b , K Ψ ~ - 1 ( a , b ) = implies that Ψ ~ a is a strong deformation retract of    Ψ ~ b K b , where K denotes the critical set of    Ψ (and also Ψ ~ ).

Lemma 14 (see [<xref ref-type="bibr" rid="B4">20</xref>]).

C * ( Ψ ~ , p 0 ) = C * ( Ψ , p 0 ) with integral coefficients.

Here and in what follows, we always assume that  Ψ has only finitely many critical points.

Lemma 15 (see [<xref ref-type="bibr" rid="B16">21</xref>]).

Let 0 be an isolated critical point of   Ψ C 2 ( E , ) , where N = ker [ Ψ ( 0 ) ] . Denote μ = dim E - < , ν = dim N < , and assume that Ψ has a local linking at 0 with respect to a direct sum decomposition E = W - W + , where W - = E - N ; that is, there exists r > 0 small such that (30) Ψ ( u ) > 0 f o r u W + , 0 < u V r , Ψ ( u ) 0 f o r u W - , u V r . Then (31) C q ( Ψ , 0 ) = δ q , k 𝔽 f o r k = μ + ν .

3. Calculation of Degree Lemma 16.

Suppose that (f1),  (f3), and (f4) hold. Then there exists r l k > 0 , such that, for all r ( 0 , r l k ] , (32) deg ( I - A , B ( 0 , r ) , 0 ) = 0 .

Proof.

Since { e j } j = 1 is an orthogonal basis of V , for u V , there exist { a j } j = 1 , such that u = j = 1 a j e j . Let (33) E - = H - N , E + = H + . Since E - is finite dimensional, we have that, for given r > 0 , there exists some ρ > 0 such that if (34) u E - , u V ρ , then (35) | u ( x ) | r 3 < r , a . e . x Ω . By (f4), for u E - with u ρ , (36) Ψ ( u ) = 1 2 Ω ( | Δ u | 2 ) d x - Ω F ( x , u ) d x 1 2 Ω ( | Δ u | 2 - μ 2 i u 2 ) d x = 1 2 Ω ( | Δ j = 1 2 i a j e j | 2 - μ 2 i j = 1 2 i a j e j ) d x 1 2 Ω ( | Δ u | 2 - m u 2 i u 2 ) d x = 1 2 Ω ( | j = 1 2 i μ j a j e j | 2 - j = 1 2 i μ 2 i a j e j ) d x 0 . For u E + with 0 < u ρ , (37) Ψ ( u ) = 1 2 Ω ( | Δ u | 2 ) d x - Ω F ( x , u ) d x > 1 2 Ω ( | Δ u | 2 - μ 2 i + 1 u 2 ) d x = 1 2 Ω ( | Δ j = 2 i + 1 a j e j | 2 - m u 2 i + 1 j = 2 i + 1 a j e j ) d x = 1 2 Ω ( | j = 2 i μ j a j e j | 2 - j = 2 i + 1 μ 2 i + 1 a j e j ) d x 0 . Thus Ψ possesses a local linking at the origin. By Lemma 15, the critical groups of Ψ at the origin satisfy (38) C μ 2 i ( Ψ , 0 ) 0 . Then there exists r l k > 0 small such that there is no other critical point in B ( 0 , r l k ) except 0 , for all r ( 0 , r l k ] , and the following can be obtained by Lemma 13 and Remark 10: (39) deg ( I - A , B ( 0 , r ) , 0 ) = 1 .

Lemma 17.

Suppose (f1) and (f2) hold. There exists r 1 ( 0 , r l k ] ( r l k is defined in Lemma 16), such that, for all r ( 0 , r 1 ] , (40) i ( A , P 2 r , P 2 ) = 0 , i ( A , - P 2 r , - P 2 ) = 0 .

Proof.

We only prove that i ( A , P 2 r , P 2 ) = 0 . By (f1),  A ( P 2 ) P 2 , it follows from the condition (f2) that there exist δ 1 > 0 and ρ 1 ( 0 , r l k ] such that (41) f ( x , t ) μ 1 ( 1 + δ 1 ) t , ( x , t ) Ω ¯ × [ 0 , ρ 1 ] .

If u = A u + ν e 1 for some ν 0 and u P 2 r , where r ( 0 , ρ 1 ] is a positive number, that is (42) Δ 2 u = f ( x , u ) + ν e 1 , in Ω . u = Δ u = 0 , on Ω , then we have from (41) that (43) Δ 2 u μ 1 ( 1 + δ 1 ) u , in Ω . u = Δ u = 0 , on Ω . Thus, (44) μ 1 Ω u ( x ) e 1 ( x ) d x μ 1 ( 1 + δ 1 ) Ω u ( x ) e 1 ( x ) d x , and this is a contradiction. Therefore, according to the property of fixed point index, we get (45) i ( A , P 2 r , P 2 ) = 0 .

Similarly, we can also show that there exists ρ 2 ( 0 , r 0 ] such that i ( A , - P 2 r , - P 2 ) = 0 for all r ( 0 , ρ 2 ] . Let r 1 = min { ρ 1 , ρ 2 } . Then the conclusion holds.

Lemma 18.

Suppose that (f1) and (f3) hold. Then there exists O + P 2 { 0 } , O - ( - P 2 { 0 } ) , such that (46) deg ( I - A , O + , 0 ) = 1 , deg ( I - A , O - , 0 ) = 1 .

Proof.

Since lim | t | f ( x , t ) / t < μ 1 uniformly for x Ω ¯ , there exist constants δ ( 0,1 ) and C 1 > 0 such that (47) f ( x , t ) μ 1 ( 1 - δ ) t + C 1 , ( x , t ) Ω ¯ × [ 0 , ) . We will first show that any solution of (1) is bounded. Suppose u 0 is a solution; then u 0 satisfy (48) Δ 2 u 0 = f ( x , u 0 ) , in Ω , u 0 = Δ u 0 = 0 , on Ω . Multiplying by u 0 , we have (49) u 0 V 2 = Ω f ( x , u 0 ) u 0 , x Ω ; by (47), it is easy to see (50) u 0 V 2 = Ω μ 1 ( 1 - δ ) u 0 2 + Ω C 1 u 0 , x Ω . Let ϵ < μ 1 δ / ( 2 C 1 ) , and using Young inequality, there exists constant C 2 = C 2 ( δ , C 1 ) such that (51) u 0 V 2 Ω μ 1 ( 1 - δ 2 ) u 0 2 + C 2 | Ω | 1 / 2 . By Poincaré inequality, there exists C 3 > 0 only dependent on Ω , such that (52) u 0 V < C 3 . By a bootstrap argument, there exists R 1 > 0 such that u 0 2 < R 1 .

Let R > R 1 , and we will show that (53) A u ν u , x P 2 B ( 0 , R ) , ν 1 . Suppose there exists ν 0 1 ,   u 0 V = R , such that A u 0 = ν 0 u 0 ; then ν 0 > 1 . By (f3), such that f ( x , t ) μ 1 ( 1 - δ ) t + C 1 , for all t 0 , (54) ν 0 u 0 = A u 0 μ 1 ( 1 - δ ) K u 0 + C 4 K 1 . Then [ 1 - μ 1 ( 1 - δ ) / K ] u 0 C 5 , where C 5 = C 1 K L ( C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) , C 0 ( Ω ¯ ) C 2 ( Ω ¯ ) ) . Then u 0 C 6 , where C 6 = ( I - K ) - 1 C 5 . Let R = C 6 ; then, for all u B ( 0 , R ) , we have (55) A u ν u x P 2 R , ν 1 . Then (56) i ( A , P 2 R , P 2 ) = 1 .

From Lemma 17, we have (57) i ( A , P 2 R P 2 r ¯ , P 2 ) = 1 .

By Lemma 18 and the strong maximum principle of second order elliptic problem, it is easy to know that if u P 2 { 0 } is a solution of (1), then u P ° 2 , and thus, by Lemma 2, u P ° 2 . Using Lemma 8, there is a bounded open O 1 P 2 ( B ( 0 , R ) B ( 0 , r ) ) , such that (58) deg ( I - A , O + , 0 ) = 1 . Similarly, there is a bounded open subset O - - ( P 2 R P ¯ 2 r ) , such that (59) deg ( I - A , O - , 0 ) = 1 .

4. Proof of Main Result Proof.

By conditions (f1) and (f3), it is easy to know that (60) ind ( I - A , ) = ind ( I - A ( ) , 0 ) = ( - 1 ) 0 = 1 ; that is, there exists R ¯ > R large enough, such that (61) deg ( I - A , B ( 0 , R ¯ ) , 0 ) = ind ( I - A , ) = 1 .

If A has no fixed point in B ( 0 , R ¯ ) ( P 2 R ( - P 2 R ) ) , then the additivity property of degree implies (62) deg ( I - A , B ( 0 , R ¯ ) , 0 ) = deg ( I - A , O + , 0 ) + deg ( I - A , O - , 0 ) + deg ( I - A , B ( 0 , r 2 ) , 0 ) . It follows that 1 = 1 + 1 + 1 . This is a contradiction. Thus (1) has at least a solution u 3 in B ( 0 , R ¯ ) ( P 2 R ( - P 2 R ) B ( 0 , ( r / 2 ) ) ) .

Conflict of Interests

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

Acknowledgments

This research was supported by the National Natural Science Foundation of China (no. 10871096) and the Project of Graduate Education Innovation of Jiangsu Province (CXLX13_367).

McKenna P. J. Walter W. Nonlinear oscillations in a suspension bridge Archive for Rational Mechanics and Analysis 1987 98 2 167 177 10.1007/BF00251232 MR866720 ZBL0676.35003 Micheletti A. M. Pistoia A. Nontrivial solutions for some fourth order semilinear elliptic problems Nonlinear Analysis: Theory, Methods & Applications 1998 34 4 509 523 10.1016/S0362-546X(97)00596-8 MR1635747 ZBL0929.35053 Zhang J. Existence results for some fourth-order nonlinear elliptic problems Nonlinear Analysis: Theory, Methods & Applications 2001 45 1 29 36 10.1016/S0362-546X(99)00328-4 MR1828065 ZBL0981.35016 Kyritsi S. T. Papageorgiou N. S. On superquadratic periodic systems with indefinite linear part Nonlinear Analysis: Theory, Methods & Applications 2010 72 2 946 954 10.1016/j.na.2009.07.035 MR2579360 ZBL1198.34066 Qian A. X. Li S. J. Multiple solutions for a fourth-order asymptotically linear elliptic problem Acta Mathematica Sinica 2006 22 4 1121 1126 10.1007/s10114-005-0665-7 MR2245242 ZBL1274.35140 Ma R. Wang H. On the existence of positive solutions of fourth-order ordinary differential equations Applicable Analysis 1995 59 1–4 225 231 10.1080/00036819508840401 MR1378037 ZBL0841.34019 Wei Z. Pang C. Positive solutions and multiplicity of fourth-order m -point boundary value problems with two parameters Nonlinear Analysis: Theory, Methods & Applications 2007 67 5 1586 1598 10.1016/j.na.2006.08.001 MR2323305 Xu J. Yang Z. Positive solutions for a fourth order p -Laplacian boundary value problem Nonlinear Analysis: Theory, Methods & Applications 2011 74 7 2612 2623 10.1016/j.na.2010.12.016 MR2776513 Zhang M. Wei Z. Existence of positive solutions for fourth-order m -point boundary value problem with variable parameters Applied Mathematics and Computation 2007 190 2 1417 1431 10.1016/j.amc.2007.02.019 MR2339733 Hofer H. Variational and topological methods in partially ordered Hilbert spaces Mathematische Annalen 1982 261 4 493 514 10.1007/BF01457453 MR682663 ZBL0488.47034 Zhang G. Q. Variational methods and sub- and supersolutions Scientia Sinica A: Mathematical, Physical, Astronomical & Technical Sciences 1983 26 12 1256 1265 MR745797 Bartsch T. Chang K.-C. Wang Z.-Q. On the Morse indices of sign changing solutions of nonlinear elliptic problems Mathematische Zeitschrift 2000 233 4 655 677 10.1007/s002090050492 MR1759266 ZBL0946.35023 Chang K.-C. Infinite-Dimensional Morse Theory and Multiple Solution Problems 1993 6 Boston, Mass, USA Birkhäuser Boston Progress in Nonlinear Differential Equations and their Applications MR1196690 Liu Z. Sun J. An elliptic problem with jumping nonlinearities Nonlinear Analysis; Theory, Methods & Applications 2005 63 8 1070 1082 10.1016/j.na.2005.03.109 MR2211582 ZBL1161.35395 Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces SIAM Review 1976 18 4 620 709 Chabrowski J. Marcos do Ó J. On some fourth-order semilinear elliptic problems in n Nonlinear Analysis: Theory, Methods & Applications 2002 49 6 861 884 10.1016/S0362-546X(01)00144-4 MR1894788 ZBL1011.35045 Xu X. Multiple sign-changing solutions for some m-point boundary-value problems Electronic Journal of Differential Equations 2004 89 1 14 MR2075428 ZBL1058.34013 Li F. Li Y. Multiple sign-changing solutions to semilinear elliptic resonant problems Nonlinear Analysis: Theory, Methods & Applications 2010 72 9-10 3820 3827 10.1016/j.na.2010.01.018 MR2606823 ZBL1185.35096 Liu Z. Positive solutions of superlinear elliptic equations Journal of Functional Analysis 1999 167 2 370 398 10.1006/jfan.1999.3446 MR1716201 ZBL0951.35051 Zhang G. Q. A variant mountain pass lemma Scientia Sinica A: Mathematical, Physical, Astronomical & Technical Sciences 1983 26 12 1241 1255 MR745796 ZBL0544.35044 Su J. Multiplicity results for asymptotically linear elliptic problems at resonance Journal of Mathematical Analysis and Applications 2003 278 2 397 408 10.1016/S0022-247X(02)00707-2 MR1974015 ZBL01915717