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

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.