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.
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 [
Xu [
We list some assumptions as follows. Suppose that the sequence of positive solutions to the equation
is
let there exists
for all
Suppose that conditions
Based on [ There exists
Pang et al. [
The main purpose of this paper is to abstract more general conditions from
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 [
Let
Let
Let
Let
Let
Let
Let
Suppose that
Let
Let
Let
Then
From condition
Since
By (
Since
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
According to the theory of elliptic partial differential equations (see [
Then
Let
For
By the proof of Lemma 4.1 in [
Let
In order to obtain multiple sign-changing solutions of (
there exists a constant number
Suppose that
From condition
It follows from
In the following, we prove that
The proof is completed.
It follows from conditions
The author declares that there is no conflict of interests regarding the publication of this paper.
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).