Electron. J. Diff. Equ., Vol. 2016 (2016), No. 135, pp. 1-16.

Multiple sign-changing solutions for Kirchhoff type problems

Cyril Joel Batkam

Abstract:
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.

Submitted December 17, 2015. Published June 7, 2016.
Math Subject Classifications: 35J60, 35A15, 35J20, 35J25.
Key Words: Kirchhoff type equation; sign-changing solution; multiple solutions; critical point theorem.

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Cyril Joel Batkam
HEC Montreal
3000 Chemin de la Cate-Sainte-Catherine
Montrel, QC, H3T 2B1, Canada
email: cyril-joel.batkam@hec.ca, cyril.joel.batkam@usherbrooke.ca

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