- Research Article
- Open Access

# Global Structure of Nodal Solutions for Second-Order *m*-Point Boundary Value Problems with Superlinear Nonlinearities

- Yulian An
^{1}Email author

**Received:**8 May 2010**Accepted:**23 September 2010**Published:**28 September 2010

## Abstract

We consider the nonlinear eigenvalue problems , , , , where , and for with and satisfies for , and , where . We investigate the global structure of nodal solutions by using the Rabinowitz's global bifurcation theorem.

## Keywords

- Banach Space
- Continuous Operator
- Global Structure
- Global Behavior
- Auxiliary Equation

## 1. Introduction

Here and for with is a positive parameter, and

In the case that the global structure of nodal solutions of nonlinear second-order -point eigenvalue problems (1.1), (1.2) have been extensively studied; see [1–5] and the references therein. However, relatively little is known about the global structure of solutions in the case that and few global results were found in the available literature when The likely reason is that the global bifurcation techniques cannot be used directly in the case. On the other hand, when -point boundary value condition (1.2) is concerned, the discussion is more difficult since the problem is nonsymmetric and the corresponding operator is disconjugate. In [6], we discussed the global structure of positive solutions of (1.1), (1.2) with However, to the best of our knowledge, there is no paper to discuss the global structure of nodal solutions of (1.1), (1.2) with

In this paper, we obtain a complete description of the global structure of nodal solutions of (1.1), (1.2) under the following assumptions:

for with

satisfies for

;

.

Then has a bounded inverse and the restriction of to that is, is a compact and continuous operator; see [1, 2, 6].

For any function if then is a simple zero of if For any integer and any define sets consisting of functions satisfying the following conditions:

(i)

(ii) has only simple zeros in and has exactly zeros in ;

(i) and

(ii) has only simple zeros in and has exactly zeros in

(iii) has a zero strictly between each two consecutive zeros of .

Remark 1.1.

Obviously, if then or The sets are open in and disjoint.

Remark 1.2.

The nodal properties of solutions of nonlinear Sturm-Liouville problems with separated boundary conditions are usually described in terms of sets similar to see [7]. However, Rynne [1] stated that are more appropriate than when the multipoint boundary condition (1.2) is considered.

We call the set of eigenvalues of (1.7) the spectrum of and denote it by . The following lemmas or similar results can be found in [1–3].

Lemma 1.3.

Let hold. The spectrum consists of a strictly increasing positive sequence of eigenvalues with corresponding eigenfunctions In addition,

(i)

(ii) for each and is strictly positive on

We can regard the inverse operator as an operator In this setting, each is a characteristic value of with algebraic multiplicity defined to be dim where denotes null-space and is the identity on

Lemma 1.4.

Let hold. For each the algebraic multiplicity of the characteristic value of is equal to 1.

Let under the product topology. As in [7], we add the points to our space Let Let denote the closure of set of those solutions of (1.1), (1.2) which belong to The main results of this paper are the following.

Theorem 1.5.

Let (A1)–(A4) hold.

(c)If then there exists a subcontinuum of with is a bounded closed interval, and approaches as

Theorem 1.6.

Let (A1)–(A4) hold.

(a)If , then (1.1), (1.2) has at least one solution in for any

(b)If , then (1.1), (1.2) has at least one solution in for any

(c)If then there exists such that (1.1), (1.2) has at least two solutions in for any .

The rest of the paper is organized as follows. Section 2 contains some preliminary propositions. In Section 3, we use the global bifurcation theorems to analyse the global behavior of the components of nodal solutions of (1.1), (1.2).

## 2. Preliminaries

Definition 2.1 (see [9]).

Lemma 2.2 (see [9]).

Each connected subset of metric space is contained in a component, and each connected component of is closed.

Lemma 2.3 (see [6]).

Assume that

(i)there exist and such that ;

(ii) , where ;

Then there exists an unbounded connected component in and .

It is easy to verify that the following lemma holds.

Lemma 2.4.

Assume that (A1)-(A2) hold. Then is completely continuous.

Lemma 2.5.

where

Proof.

The proof is similar to that of Lemma in [6]; we omit it.

Lemma 2.6.

Proof.

which implies that is bounded whenever is bounded.

## 3. Proof of the Main Results

Lemma 3.1 (see [1, Proposition ]).

Let (A1), (A2) hold. If is a nontrivial solution of (3.4), (3.5), then for some .

as a bifurcation problem from the trivial solution

Further we note that for near in

The results of Rabinowitz [8] for (3.8) can be stated as follows. For each integer , there exists a continuum of solutions of (3.8) joining to infinity in Moreover,

Proof of Theorem 1.5.

Let us verify that satisfies all of the conditions of Lemma 2.3.

and accordingly, holds. can be deduced directly from the Arzela-Ascoli Theorem and the definition of . Therefore, the superior limit of , contains an unbounded connected component with .

has a unique solution on for every and . Therefore, any nontrivial solution of (1.1), (1.2) has only simple zeros in and . Meanwhile, (A1) implies that [1, proposition 4.1]. Since , we conclude that . Moreover, by (1.1) and (1.2).

We divide the proof into three cases.

Case 1 ( ).

In this case, we show that .

Case 2 ( ).

In this case, we can show easily that joins with by using the same method used to prove Theorem in [2].

Case 3 ( ).

In this case, we show that joins with .

Taking subsequences again if necessary, we still denote such that . If , all the following proofs are similar.

denote the zeros of in . Then, after taking a subsequence if necessary, . Clearly, . Set . We can choose at least one subinterval which is of length at least for some . Then, for this if is large enough. Put .

We see that consists of a sequence of positive and negative bumps, together with a truncated bump at the right end of the interval with the following properties (ignoring the truncated bump) (see, [1]):

(i)all the positive (resp., negative) bumps have the same shape (the shapes of the positive and negative bumps may be different);

(ii)each bump contains a single zero of , and there is exactly one zero of between consecutive zeros of ;

(iii)all the positive (negative) bumps attain the same maximum (minimum) value.

Now, we show that .

From (3.28) we obtain that must change its sign on if is large enough. This is a contradiction. Therefore .

Proof of Theorem 1.6.

and are immediate consequence of Theorem 1.5 and , respectively.

where

By Lemma 2.6 and Theorem 1.5, it follows that is also an unbounded component joining and in . Thus, (3.40) implies that for (1.1), (1.2) has at least two solutions in .

## Declarations

### Acknowledgments

The author is very grateful to the anonymous referees for their valuable suggestions. This paper was supported by NSFC (no.10671158), 11YZ225, YJ2009-16 (no.A06/1020K096019).

## Authors’ Affiliations

## References

- Rynne BP: Spectral properties and nodal solutions for second-order,
*m*-point, boundary value problems.*Nonlinear Analysis: Theory, Methods & Applications*2007, 67(12):3318-3327. 10.1016/j.na.2006.10.014View ArticleMathSciNetGoogle Scholar - Ma R, O'Regan D: Nodal solutions for second-order
*m*-point boundary value problems with nonlinearities across several eigenvalues.*Nonlinear Analysis: Theory, Methods & Applications*2006, 64(7):1562-1577. 10.1016/j.na.2005.07.007View ArticleMathSciNetGoogle Scholar - Xu X: Multiple sign-changing solutions for some m-point boundary-value problems.
*Electronic Journal of Differential Equations*2004, 2004(89):1-14.Google Scholar - Jingxian S, Xian X, O'Regan D: Nodal solutions for
*m*-point boundary value problems using bifurcation methods.*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(10):3034-3046. 10.1016/j.na.2007.02.043View ArticleMathSciNetGoogle Scholar - An Y, Ma R: Global behavior of the components for the second order
*m*-point boundary value problems.*Boundary Value Problems*2008, 2008:-10.Google Scholar - Ma R, An Y: Global structure of positive solutions for superlinear second order
*m*-point boundary value problems.*Topological Methods in Nonlinear Analysis*2009, 34(2):279-290.MathSciNetGoogle Scholar - Ma R, Thompson B: Nodal solutions for nonlinear eigenvalue problems.
*Nonlinear Analysis: Theory, Methods & Applications*2004, 59(5):707-718.View ArticleMathSciNetGoogle Scholar - Rabinowitz PH: Some global results for nonlinear eigenvalue problems.
*Journal of Functional Analysis*1971, 7: 487-513. 10.1016/0022-1236(71)90030-9View ArticleMathSciNetGoogle Scholar - Whyburn GT:
*Topological Analysis, Princeton Mathematical Series. No. 23*. Princeton University Press, Princeton, NJ, USA; 1958:xii+119.Google Scholar - Naito Y, Tanaka S: Sharp conditions for the existence of sign-changing solutions to equations involving the one-dimensional
*p*-Laplacian.*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(9):3070-3083. 10.1016/j.na.2007.09.002View ArticleMathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.