Infinitely many solutions for Kirchhoff equations with indefinite potential

Abstract

We obtain a sequence of solutions converging to zero for the Kirchhoff equation $$-\left(1+{\int }_{\Omega }{\left|\nabla u\right|}^2\right)\Delta u+V(x)u=f(u)\text{,}u\in H_0^1(\Omega )$$ via truncating technique and a variant of Clark's theorem due to Liu--Wang, where $\Omega$ is a bounded smooth domain $\Omega \subset {\mathbb{R}}^N$. Similar result for Schrödinger-Poisson system on a bounded smooth domain $\Omega \subset {\mathbb{R}}^3$ is also presented.

7 pages