Normalized solutions for nonautonomous Schr\"{o}dinger-Poisson equations

Abstract

In this paper, we study the existence of normalized solutions for the nonautonomous Schr\"{o}dinger-Poisson equations \begin{equation}\nonumber -\Delta u+\lambda u +\left(\vert x \vert ^{-1} * \vert u \vert ^{2} \right) u=A(x)|u|^{p-2}u,\quad \text{in}~\R^3, \end{equation} where $\lambda\in\R$, $A \in L^\infty(\R^3)$ satisfies some mild conditions. Due to the nonconstant potential $A$, we use Pohozaev manifold to recover the compactness for a minimizing sequence. For $p\in (2,3)$, $p\in(3,\frac{10}{3})$ and $p\in(\frac{10}{3}, 6)$, we adopt different analytical techniques to overcome the difficulties due to the presence of three terms in the corresponding energy functional which scale differently, respectively.