Multiple sign-changing solutions to a class of Kirchhoff type problems

Abstract

This paper is concerned with the existence of sign-changing solutions to non local Kirchhoff type problems of the form \begin{equation}\label{s}\tag{S} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u)\, \text{ in }\Omega,\quad\quad u=0 \text{ on }\partial\Omega, \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N=1,2,3$) with smooth boundary, $a>0$, $b>0$, and $f:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is a continuous function. We give a positive answer to a long standing question concerning the existence of more than two sign-changing solutions to \eqref{s}. More precisely, we show in this paper that if $f$ is globally 3-superlinear, subcritical and odd with respect to the second variable, then \eqref{s} possesses an unbounded sequence of sign-changing solutions. Our approach is variational and relies on a new sign-changing version of the symmetric mountain pass theorem established in this paper.