Sign-Changing Solutions for Nonlinear Operator Equations

Abstract

The existence of six solutions for nonlinear operator equations is obtained by using the topological degree and fixed point index theory. These six solutions are all nonzero. Two of them are positive, the other two are negative, and the fifth and sixth ones are both sign-changing solutions. Furthermore, the theoretical results are applied to elliptic partial differential equations.

1. Introduction

In recent years, motivated by some ecological problems, much attention has been attached to the existence of sign-changing solutions for nonlinear partial differential equations (see [1–4] and the references therein). We note that the proofs of main results in [1–4] depend upon critical point theory. However, some concrete nonlinear problems have no variational structures [5]. To overcome this difficulty, in [6], Zhang studied the existence of sign-changing solution for nonlinear operator equations by using the cone theory and combining uniformly positive condition.

Xu [7] studied multiple sign-changing solutions to the following -point boundary value problems: where , , .

We list some assumptions as follows.(A₁)Suppose that the sequence of positive solutions to the equation  is ;(A₂), is a continuous function, , and for all ;(A₃)let and . There exist positive integers and such that (A₄)there exists such that for all with .

Theorem 1 (see [7]). Suppose that conditions are satisfied. Then the problem (1) has at least two sign-changing solutions. Moreover, the problem (1) also has at least two positive solutions and two negative solutions.

Based on [7], many authors studied the sign-changing solutions of differential and difference equations. For example, Yang [8] considered the existence of multiple sign-changing solutions for the problem (1). Compared with Theorem 1, Yang employed the following assumption which is different from .(A′₄)There exists such that Pang et al. [9] investigated multiple sign-changing solutions of fourth-order differential equation boundary value problems. Moreover, Wei and Pang [10] established the existence theorem of multiple sign-changing solutions for fourth-order boundary value problems. Y. Li and F. Li [11] studied two sign-changing solutions of a class of second-order integral boundary value problems by computing the eigenvalues and the algebraic multiplicities of the corresponding linear problems. He et al. [12] discussed the existence of sign-changing solutions for a class of discrete boundary value problems, and a concrete example was also given. Very recently, Yang [13] investigated the following discrete fourth Neumann boundary value problems The author employed similar conditions with and obtained a similar result to Theorem 1 (see Theorem  5.1 in [13]).

The main purpose of this paper is to abstract more general conditions from of Theorem 1, obtain the existence theorem of sign-changing solutions for general operator equations, and, then, apply the abstract result obtained in this paper to nonlinear elliptic partial differential equations.

2. Preliminaries and Some Lemmas

For the discussion of the following sections, we state here preliminary definitions and known results on cones, partial orderings, and topological degree theory, which can be found in [14–18].

Let be a real Banach space. Given a cone , we define a partial ordering with respect to by if and only if . A cone is said to be normal if there exists a constant such that implies ; the smallest is called the normal constant of . is called solid if it contains interior, that is, . If and , we write ; if cone is solid and , we write . is reproducing if and total if . Let be a bounded linear operator. is said to be positive if . An operator is strongly increasing; implies . If is a linear operator, is strongly increasing which implies is strongly positive. A fixed point of operator is said to be a sign-changing fixed point if . If satisfies , where is some real number, then is called an eigenvalue of and is called an eigenfunction belonging to the eigenvalue .

Definition 2 (see [16]). Let , be real Banach spaces and let contain the outside of a ball , and . The operator is called asymptotically linear if there is a bounded linear operator such that The operator involved in the definition of an asymptotically linear operator is uniquely determined. It is called the derivative of at infinity and is denoted by .

Definition 3 (see [16, 18]). Let be a retract of , and let be a relatively bounded open set of . Suppose that is completely continuous and has no fixed point on . Let the positive integer be defined by where is an arbitrary retraction, and is a large enough positive number such that . Then is called the fixed point index of on with respect to .

Lemma 4 (see [14, 18]). Let be a reproducing cone and let be a positive completely continuous linear operator, with , where denotes the spectral radius of . Then has a positive eigenfunction in corresponding to the eigenvalue .

Lemma 5 (see [18]). Let be a cone in and let be a bounded open set of , and . Assume that is a condensing operator. If for every , then .

Lemma 6 (see [19]). Let be a normal and total cone in , and let be a completely continuous increasing operator. Then the following assertions hold(a), is Fréchet differentiable at . If , , and there exist and such that , then there exists such that for all , where ;(b) is an asymptotically linear operator. If , , and there exist and such that , then there exists such that for all .

Lemma 7 (see [18]). Let be an open set of , be completely continuous, , and . Assume that is Fréchet differentiable at and is not an eigenvalue of , then is an isolated fixed point, and where is the sum of algebraic multiplicities of the real eigenvalues of in .

Lemma 8 (see [18]). Suppose that is a completely continuous and asymptotically linear operator. If is not an eigenvalue of the linear operator , then there exists such that for all , where is the sum of the algebraic multiplicities of the real eigenvalues of in .

Lemma 9 (see [9]). Let be a solid cone in , be a relatively bounded open subset of , and be a completely continuous operator. If any fixed point of in is an interior point of , then there exists an open subset of () such that

Lemma 10 (see [18]). Let be a bounded open set of , and let . Assume that is a condensing operator. If then .

3. Multiple Sign-Changing Solutions for Nonlinear Operator Equations

Theorem 11. Let be a normal solid cone in , be a completely continuous operator, , , and . Suppose that (H₁) exists and is an increasing operator; ; is not an eigenvalue of , and the sum of algebraic multiplicities for the real eigenvalues of in is an even number;(H₂) exists and is an increasing operator; ; is not an eigenvalue of , and the sum of algebraic multiplicities for the real eigenvalues of in is an even number;(H₃) .Then has at least two sign-changing fixed points, two positive fixed points, and two negative fixed points.

Proof. From condition , we obtain that there exists such that for all . By Lemma 5, there exists such that where .
Since , , and , which together with imply that and . According to Lemma 4, we know that there exists such that . Since 1 is not an eigenvalue of , and . By Lemma 6, there exists such that for all . Similarly, we get that In the same sense, we know that there exists such that for all . Further, combining Lemmas 7 and 8 with and , we get that there exist and such that
By (15)–(17), we have It follows from (13), (20), (22), and the additivity property of fixed point index that Hence has at least two fixed points and in and , respectively. It is obvious that and are both positive. Moreover, it follows from (14), (21), (23), and the additivity property of fixed point index that Consequently, also has at least two fixed points and in and , respectively. Evidently, and are both negative.
Since , . Let By Lemma 9 and (24)–(25), we get that there exist open sets () of such that By Lemma 10, we have According to (32), (28), (30), (18), and the additivity property of Leray-Schauder degree, we obtain which yields that has at least one fixed point in , and then is a sign-changing fixed point. It follows from (19), (29), (31), (32), and the additivity property of Leray-Schauder degree that we have which implies that has at least one fixed point in , and then is also a sign-changing fixed point. The proof is completed.

4. Example

The main purpose of this section is to apply our theorem to nonlinear differential equations.

We consider the following boundary value problem for elliptic partial differential equations where is a bounded open domain in , , and ; is continuous; is an uniformly elliptic operator; that is, and there exists a constant number such that for all . Consider which is a boundary operator, where is a vector field on of satisfying ( denotes the outer unit normal vector on ) and , and assume that one of the following cases holds: (i) and ;(ii) and ;(iii) and .

According to the theory of elliptic partial differential equations (see [20, 21]), we know that for each , the linear boundary value problem has a unique solution . Define the operator by

Then is a linear completely continuous operator and has an unbounded sequence of eigenvalues: and the spectral radius .

Let Then is an ordered Banach space with the norm and is a normal solid cone in and .

For , define Nemytskii operator by Clearly, is continuous. Let . Then is completely continuous.

By the proof of Lemma  4.1 in [22], we have that .

Let be the solution of the following boundary value problem

In order to obtain multiple sign-changing solutions of (35), we give the following assumptions.(E₁) and ; , , , and ;(E₂) uniformly for , and , where is an even number;(E₃) uniformly for , and , where is an even number;(E₄) there exists a constant number such that uniformly for .

Theorem 12. Suppose that are satisfied. Then the problem (35) has at least two sign-changing solutions. Moreover, problem (35) has at least two positive solutions and two negative solutions.

Proof. From condition , we know that Copy the proof proceed of Theorem  3.4 in [23], we have that , and , where where is the first normalized eigenfunction of corresponding to its first eigenvalue .
It follows from and that conditions and of Theorem 11 hold.
In the following, we prove that of Theorem 11 is satisfied. It follows from that there exists such that That is, Thus Therefore,
The proof is completed.

Remark 13. It follows from conditions and that . We should point out that the initial ideas of condition and the general one are motivated by [24].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author was supported financially by the National Natural Science Foundation of China, Tianyuan Foundation (11226119), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi, the Youth Science Foundation of Shanxi Province (2013021002-1), and Shandong Provincial Natural Science Foundation, China (ZR2012AQ024).