## Boundary Value Problems

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# Multiple Nodal Solutions for Some Fourth-Order Boundary Value Problems via Admissible Invariant Sets

Boundary Value Problems20082008:403761

DOI: 10.1155/2008/403761

Accepted: 16 September 2008

Published: 9 October 2008

## Abstract

Existence and multiplicity results for nodal solutions are obtained for the fourth-order boundary value problem (BVP) , , , where is continuous. The critical point theory and admissible invariant sets are employed to discuss this problem.

## 1. Introduction

In this paper, we consider the existence of nodal solutions to the semilinear fourth-order equation:
(1.1)
(1.2)

where is continuous.

Owning to the importance of higher-order differential equations in physics, the existence and multiplicity of the solutions to such problems have been studied by many authors. They obtained the existence of solutions by the cone expansion or compression fixed point theorem [16]; sub-sup solution method [79]; critical point theory [1013]; Morse theory [14, 15]; and eta [16, 17]. There are also papers which study nodal solutions for elliptic equations [18, 19]. In particular, in [20], Han and Li obtained multiple positive, negative, and sign-changing solutions by combining the critical point theory and the method of sub-sup solutions for the (BVP) (1.2). The main result is as follows:

there exist a strict subsolution and a strict supersolution of (BVP) (1.2) with , , and ;

is strictly increasing in ;

is locally Lipschitz continuous in ;

there exist and such that for all and

Theorem 1.1 (see [20]).

Assume that hold. Then, (BVP) (1.2) has at least four solutions.

Motivated by their ideas, we cannot help wondering if there are no strict subsolution and supersolution of (BVP) (1.2), can we still get the nodal solutions just by critical point theory? In this paper, we will use the admissible invariant sets and critical point theory to settle this problem. But we should point out that in all theorems of our paper, the nonlinearity is assumed to be odd in , while no such symmetry is required in [20].

The paper is organized as follows: in Section 2, we give some preliminaries, including the critical point theorems which will be used in our main results and some concepts concerning the partially ordered Banach space. The main results and proofs are established in Section 3.

## 2. Preliminaries

Let be a Hilbert space and a Banach space densely embedded in . Assume that has a closed convex cone and that has interior points in , that is, with the interior and the boundary of in .

Let and for . We use the following notation: , , , for . Let and denote the norms in and , respectively.

Lemma 2.1 (see [21]).

Assume is a Hilbert space, and is a closed convex set of , , and . Then, there exists a pseudogradient vector field for , and . Furthermore, if is even, , then is odd.

Consider the pseudogradient flow on associated with the vector field ,
(2.1)

We see that is odd in , if is odd in . Since for and the Brezis-Martin theorem [22] implies that for .

Definition 2.2 (see [21, 23]).

With the flow , a subset is called an invariant set if for .

Let us assume that

(F) , for , is continuous.

Under condition , we have for and is continuous in .

Definition 2.3 (see [21]).

Let be an invariant set under . is said to be an admissible invariant set for if (a) is the closure of an open set in , that is, ; (b) if for some and in as for some , then in ; (c) if such that in , then in ; (d) for any , for .

Lemma 2.4 (see [24]).

Let and hold. Assume is even, bounded from below, and satisfies (PS) condition. Assume that the positive cone is an admissible invariant set for and for all . Suppose there is a linear subspace with , such that for some , where Then, has at least pairs of critical points with negative critical values. More precisely,

(i)if , has at least one pair of critical points in and at least pairs of critical points in

(ii)if has at least one pair of critical points in and at least pairs of critical points in

Lemma 2.5 (see [21]).

Let and hold. Assume is even, , and satisfies (PS) condition. Assume that the positive cone is an admissible invariant set for and for all . Suppose there exist linear subspaces and with , ( , resp.), , such that for some , and . Then, has at least ( , resp.) pairs of critical points in with negative critical values.

Lemma 2.6 (see [21]).

Let and hold. Assume is even, and satisfies (PS) condition. Assume that the positive cone is an admissible invariant set for and for all . Suppose there exist linear subspaces and with , , , such that for some , and . Then for ( , resp.), has at least ( , resp.) pairs of critical points in with positive critical values.

Lemma 2.7 (see [21, 25]).

Assume is even, , satisfies and condition for . Assume that is an admissible invariant set for , for all . , where are finite-dimensional subspaces of , and for each , let and Assume for each there exist such that , where , as . Then, has a sequence of critical points such that as , provided for large .

Next, we need some basic concepts of ordered Banach spaces.

Definition 2.8.

An ordered real Banach space is a pair , where is a real Banach space and a closed convex subset of such that and . The partial order on is given by the cone . For , we write
(2.2)
If has nonempty interior, then it is called a solid cone. If every ordered interval is bounded, then is called a normal cone. An operator is called order preserving (in the literature sometimes increasing) if
(2.3)
strictly order preserving if
(2.4)
and strongly order preserving if
(2.5)

## 3. Main Results

In this section, we will employ the abstract results in Section 2 to establish some existence theorems on sign-changing solutions of (BVP) (1.2). Firstly, we give some lemmas to change (BVP) (1.2) to a variational problem. Let be the usual real Banach space with the norm for all . We can easily verify that
(3.1)
is also a Banach space with respect to . Let
(3.2)
then is a normal solid cone in and
(3.3)

By , we denote the usual real Hilbert space with the inner product for all

It is well known that the solution of (BVP) (1.2) in is equivalent to the solution of the following integral equation in :
(3.4)
where is the Green's function of the linear boundary value problem for all subject to that is,
(3.5)
Define operators by
(3.6)
Since , (3.4) is equivalent to the following operator equation in :
(3.7)

Remark 3.1.

It is easy to see that
1. (i)

is nonnegative continuous;

2. (ii)

;

3. (iii)

is bounded and continuous.

Lemma 3.2 (see [20]).

is a linear completely continuous operator and also a linear completely continuous operator from In addition, is strongly order-preserving.

From the definition of , we can obtain that for all with . Therefore, for all with . It is well known that all eigenvalues of are
(3.8)
which have the corresponding orthonormal eigenfunctions
(3.9)

and .

Lemma 3.3 (see [10]).

The operator equation
(3.10)
has a solution in if and only if the operator equation
(3.11)

has a solution in .

The uniqueness of the solution for these two above equations is also equivalent.

Remark 3.4.

From the proof of Lemma 3.3 [10], it is very clear if is a solution for (3.11), then is a solution for (3.7). Furthermore, if is a solution for (3.11), then is a solution for (3.7) with the same sign, which follows from Lemma 3.2.

Lemma 3.5 (see [10]).

Let , . Then,

(i) is Fréchet differentiable on and for all

(ii) is Fréchet differentiable on and for all

Choose and to be our Hilbert space and Banach space, respectively. Define a functional :
(3.12)
Then, according to Lemma 3.5, we have
(3.13)

Hence, Lemma 3.3 implies that the operator equation has a solution in if and only if the functional has a critical point in . Thus, (BVP) (1.2) has been transformed into a variational problem.

We refer the following assumption:

is continuous and increasing in .

Lemma 3.6.

Under , is satisfied, and is strongly order-preserving.

Proof.

The proof is similar to [20], and we omit it here.

Lemma 3.7.

Under , is an admissible invariant set for .

Proof.

We know that is strongly order-preserving, so does given in Lemma 2.1. The Brezis-Martin theory implies that and are invariant sets under the negative pesudogradient flow of . Requirement (a) is satisfied automatically. For (d), we note that for all , we have , similar to the proof in [23], . To prove (b), let for some , so , let be a sequence such that in for some , then in . For (c), if , then , if in , for , then and , so in , and the proof is completed.

Lemma 3.8 (see [15]).

Any bounded sequence such that as has a convergent subsequence.

Next, we make more assumptions:

(f 2 ) uniformly for ;

(f 3 ) , uniformly for and some ;

(f 4 ) is odd in .

Theorem 3.9 (sublinear nonlinearity).

Under , (BVP) (1.2) has at least one pair of one-sign solutions , , and at least pairs of nodal solutions for .

Proof.

It is easy to see that and holds. is an admissible invariant set for , and for . Also, is even, . By , there exist , such that for all , then
(3.14)

So is coercive, bounded from below, and satisfies (PS) condition.

Take ; from , there exist , such that , , choose , then , and
(3.15)

so for small. Result follows from Lemma 2.4.

Next, we consider an asymptotically linear problem:

(f 5 ) uniformly for ;

(f 6 ) , uniformly for .

Theorem 3.10 (asymptotically linear case).

Under , , , and , (BVP) (1.2) has at least pairs of nodal solutions provided or . Here, , if ; and , if .

Proof.

Take and such that for , . Now let be a (PS) sequence for . Writing with , , and taking inner product of and , we see that
(3.16)

So is bounded, where . Then, satisfies the (PS) condition.

If , let , and , then , and .

From , we know that there exist and such that
(3.17)
Then, for , , we can obtain, when ,
(3.18)

So, choose , then .

From , we can get there exist , such that
(3.19)
Then, when , we have
(3.20)

Choose large enough such that , and , result follows from Lemma 2.6.

If , let , , then , . From (3.17), when ,
(3.21)
When , we know from (3.19),
(3.22)

which means , then result follows from Lemma 2.5.

Next, we consider a superlinear problem. Assume that

there is such that for large;

there are , such that for large.

Theorem 3.11 (superlinear nonlinearity).

Under , , , and , (BVP) (1.2) has infinitely many nodal solutions.

Proof.

From condition by the standard argument, satisfies condition for every . Let . From , we obtain for all . Define , it is very clear and , so and . So if ,
(3.23)
Choosing , we obtain, if and ,
(3.24)
Let . From , after integrating, we obtain the existence of such that for . Hence, we have for and is constant. Therefore, when ,
(3.25)
Noting , choose large enough, such that , and
(3.26)

Result follows from Lemma 2.7.

Remark 3.12.

If there exist no strict supsolution and supersolution required in [20], just only using the functional to get the critical point [10, 11], then we just know that (BVP) (1.2) has solutions, even we can know the sign of the critical point of the functional because is not strongly order-preserving in . In our paper, using admissible invariant sets in , we can settle the problem.

## Declarations

### Acknowledgments

The authors are grateful to the referees for their useful suggestions which have improved the writing of the paper. Jihui Zhang thanks Z. Zhang and the members of AMSS very much for their hospitality and invitation to visit the Academy of Mathematics and Systems Sciences (AMSS), Academia Sinica, in January 2008. The authors also would like to thank Professor D. Cao, Professor S. Li, Professor Y. Ding, and Professor H. Yin for their help and many valuable discussions. This research was supported by the NNSF of China (Grant no.10871096), Foundation of Major Project of Science and Technology of Chinese Education Ministry, SRFDP of Higher Education, and NSF of Education Committee of Jiangsu Province. Zhitao Zhang was supported by NNSF of China (Grant no.10671195).

## Authors’ Affiliations

(1)
Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University
(2)
School of Science, Jiangnan University
(3)
Academy of Mathematics and System Sciences, Institute of Mathematics, the Chinese Academy of Sciences

## References

1. Davis JM, Eloe PW, Henderson J: Triple positive solutions and dependence on higher order derivatives. Journal of Mathematical Analysis and Applications 1999,237(2):710-720. 10.1006/jmaa.1999.6500
2. Davis JM, Henderson J, Wong PJY: General Lidstone problems: multiplicity and symmetry of solutions. Journal of Mathematical Analysis and Applications 2000,251(2):527-548. 10.1006/jmaa.2000.7028
3. Bai Z, Wang H: On positive solutions of some nonlinear fourth-order beam equations. Journal of Mathematical Analysis and Applications 2002,270(2):357-368. 10.1016/S0022-247X(02)00071-9
4. Graef JR, Qian C, Yang B: Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations. Proceedings of the American Mathematical Society 2003,131(2):577-585. 10.1090/S0002-9939-02-06579-6
5. Li Y: Positive solutions of fourth-order periodic boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2003,54(6):1069-1078. 10.1016/S0362-546X(03)00127-5
6. Yao Q: Positive solutions for eigenvalue problems of fourth-order elastic beam equations. Applied Mathematics Letters 2004,17(2):237-243. 10.1016/S0893-9659(04)90037-7
7. Ruyun M, Jihui Z, Shengmao F: The method of lower and upper solutions for fourth-order two-point boundary value problems. Journal of Mathematical Analysis and Applications 1997,215(2):415-422. 10.1006/jmaa.1997.5639
8. Bai Z: The method of lower and upper solutions for a bending of an elastic beam equation. Journal of Mathematical Analysis and Applications 2000,248(1):195-202. 10.1006/jmaa.2000.6887
9. Charkrit S, Kananthai A: Existence of solutions for some higher order boundary value problems. Journal of Mathematical Analysis and Applications 2007,329(2):830-850. 10.1016/j.jmaa.2006.06.092
10. Li F, Zhang Q, Liang Z: Existence and multiplicity of solutions of a kind of fourth-order boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2005,62(5):803-816. 10.1016/j.na.2005.03.054
11. Liu X-L, Li W-T: Existence and multiplicity of solutions for fourth-order boundary value problems with parameters. Journal of Mathematical Analysis and Applications 2007,327(1):362-375. 10.1016/j.jmaa.2006.04.021
12. Li F, Li Y, Liang Z: Existence of solutions to nonlinear Hammerstein integral equations and applications. Journal of Mathematical Analysis and Applications 2006,323(1):209-227. 10.1016/j.jmaa.2005.10.014
13. Li F, Li Y, Liang Z:Existence and multiplicity of solutions to th-order ordinary differential equations. Journal of Mathematical Analysis and Applications 2007,331(2):958-977. 10.1016/j.jmaa.2006.09.025
14. Han G, Xu Z: Multiple solutions of some nonlinear fourth-order beam equations. Nonlinear Analysis: Theory, Methods & Applications 2008,68(12):3646-3656. 10.1016/j.na.2007.04.007
15. Yang Y, Zhang J: Existence of solutions for some fourth-order boundary value problems with parameters. Nonlinear Analysis: Theory, Methods & Applications 2008,69(4):1364-1375. 10.1016/j.na.2007.06.035
16. Agarwal RP, Kiguradze I: Two-point boundary value problems for higher-order linear differential equations with strong singularities. Boundary Value Problems 2006, 2006:-32.Google Scholar
17. Perera K, Zhang Z:Multiple positive solutions of singular -Laplacian problems by variational methods. Boundary Value Problems 2005,2005(3):377-382. 10.1155/BVP.2005.377
18. Cao D, Noussair ES:Multiplicity of positive and nodal solutions for nonlinear elliptic problems in . Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1996,13(5):567-588.
19. Cao D, Noussair ES, Yan S: Solutions with multiple peaks for nonlinear elliptic equations. Proceedings of the Royal Society of Edinburgh. Section A 1999,129(2):235-264. 10.1017/S030821050002134X
20. Han G, Li F: Multiple solutions of some fourth-order boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2007,66(11):2591-2603. 10.1016/j.na.2006.03.042
21. Qian A, Li S: Multiple nodal solutions for elliptic equations. Nonlinear Analysis: Theory, Methods & Applications 2004,57(4):615-632. 10.1016/j.na.2004.03.010
22. Chang KC: Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston, Mass, USA; 1993:x+312.
23. Liu Z, Sun J: Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. Journal of Differential Equations 2001,172(2):257-299. 10.1006/jdeq.2000.3867
24. Li S, Wang Z-Q: Ljusternik-Schnirelman theory in partially ordered Hilbert spaces. Transactions of the American Mathematical Society 2002,354(8):3207-3227. 10.1090/S0002-9947-02-03031-3
25. Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics. Volume 65. American Mathematical Society, Washington, DC, USA; 1986:viii+100.