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_Ω|\nabla u|^2dx\Big)Δu=f(x,u)\, \text{ in }Ω,\quad\quad u=0 \text{ on }\partialΩ, \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.

15 pages

References in corpus (2)

• Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system
• Ground state solution of a nonlocal boundary-value problem

Cited by in corpus (1)

• Schrodinger-Kirchhoff-Poisson type systems