Infinitely many sign-changing solutions to Kirchhoff-type equations

Abstract

In this paper we study the existence of multiple sign-changing solutions for the following nonlocal Kirchhoff-type boundary value problem:

$$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\Omega }|\nabla u|^2{ dx}\right) \triangle {u}=\lambda |u|^{p-1}u,&{}\quad \text{ in }\quad \Omega ,\\ u=0,&{} \quad \text{ on }\quad \partial \Omega . \\ \end{array}\right. \end{aligned}$$

Using a new method, we prove that this problem has infinitely many sign-changing solutions and has a least energy sign-changing solution for $p\in (3,5)$. Few existence results of multiple sign-changing solutions are available in the literature. This new method is that, by choosing some suitable subsets which separate the action functional and on which the functional is bounded, so that we can use genus and the method of invariant sets of descending flow to construct the minimax values of the functional. Our work generalize some results in literature.