Multiple sign-changing solutions for Kirchhoff type problems

This article concerns the existence of sign-changing solutions to nonlocal Kirchhoff type problems of the form $$ -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \text{ in }\Omega,\quad u=0 \text{ on }\partial\Omega, $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N=1,2,3$)...

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Bibliographic Details
Main Author: Cyril Joel Batkam
Format: Article
Language:English
Published: Texas State University 2016-06-01
Series:Electronic Journal of Differential Equations
Subjects:
Online Access:http://ejde.math.txstate.edu/Volumes/2016/135/abstr.html
Description
Summary:This article concerns the existence of sign-changing solutions to nonlocal Kirchhoff type problems of the form $$ -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \text{ in }\Omega,\quad u=0 \text{ on }\partial\Omega, $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N=1,2,3$) with smooth boundary, $a>0$, $b\geq0$, and $f:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is a continuous function. We first establish a new sign-changing version of the symmetric mountain pass theorem and then apply it to prove the existence of a sequence of sign-changing solutions with higher and higher energy.
ISSN:1072-6691