On the existence and multiplicity of solutions to a Schrödinger–Kirchhoff–Poisson system with superlinear terms

Abstract

This paper is concerned with the existence and multiplicity of solutions to the following Schrödinger–Kirchhoff–Poisson system $\left(\begin{array}{c}-(a+b{\int }_{\Omega }|\nabla u{|}^2)\Delta u+K\left(x\right)\varphi u=f(x,u),x\in \Omega ,\\ -\Delta \varphi =K\left(x\right)u^2,x\in \Omega ,\\ u=\varphi =0,x\in \partial \Omega ,\end{array}\right)$where $a\ge 0$ and $b>0$ and $\Omega$ is a bounded smooth domain of $R^3$. Under certain assumptions of nonnegative density charge $K\left(x\right)$ and superlinear term $f(x,u)$, we establish the existence of infinitely many nontrivial solutions by means of the symmetric mountain pass theorem.