Schrödinger-Kirchhoff-Poisson type systems

Abstract

In this article, we are concerned with the boundary value problem\begin{equation} \left\{ \begin{array}{ll} \displaystyle -\left(a+b\int_{\Omega}|\nabla u|^{2}\right)\Delta u +\phi u= f(x, u) &\text{in }\Omega \hbox{} \\ -\Delta \phi= u^{2} &\text{in }\Omega \hbox{} \\ u=\phi=0&\text{on }\partial\Omega, \hbox{} \end{array} \right.\end{equation}where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$ ($N=1,2$ or $3$), $a>0$, $b\geq0$, and $f:\overline{\Omega}\times \mathbb{R}\to\mathbb{R}$ is a continuous function which is globally $3$-superlinear. By using some variants of the mountain pass theorem established in this paper, we show that this problem has at least three solutions: one positive, one negative, and one which changes its sign. Furthermore, in case $f$ is odd with respect to $u$ we obtain an unbounded sequence of sign-changing solutions.