Infinitely many solutions for Kirchhoff equations with indefinite potential

TL;DR

This paper proves the existence of infinitely many solutions converging to zero for a class of Kirchhoff equations with indefinite potential, using variational methods and truncation techniques, and extends results to Schrödinger-Poisson systems.

Contribution

It introduces a novel approach combining truncation and Clark's theorem variants to establish multiple solutions for Kirchhoff and Schrödinger-Poisson equations with indefinite potentials.

Findings

Sequence of solutions converging to zero for Kirchhoff equations.

Extension of results to Schrödinger-Poisson systems.

Application of variational methods and truncation techniques.

Abstract

We obtain a sequence of solutions converging to zero for the Kirchhoff equation $$-\left( 1+\int_{\Omega}\left\vert \nabla u\right\vert^2\right) \Delta u+V(x)u=f(u)\text{,\qquad}u\in H_{0}^{1}(\Omega)$$ 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 Schr\"{o}dinger-Poisson system on a bounded smooth domain $\Omega\subset\mathbb{R}^3$ is also presented.