Bound state nodal solutions for the non-autonomous Schr\"{o}dinger--Poisson system in $\mathbb{R}^{3}$

Abstract

In this paper, we study the existence of nodal solutions for the non-autonomous Schr\"{o}dinger--Poisson system: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda K(x) \phi u=f(x) |u|^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2} & \text{ in }\mathbb{R}^{3},% \end{array}% \right. \end{equation*}% where $\lambda >0$ is a parameter and $2<p<4$. Under some proper assumptions on the nonnegative functions $K(x)$ and $f(x)$, but not requiring any symmetry property, when $\lambda$ is sufficiently small, we find a bounded nodal solution for the above problem by proposing a new approach, which changes sign exactly once in $\mathbb{R}^{3}$. In particular, the existence of a least energy nodal solution is concerned as well.