On a class of mixed Choquard-Schrödinger-Poisson system

Abstract

We study the system $$ \left\{ -Δu+u+K(x) ϕ|u|^{q-2}u&=(I_α*|u|^p)|u|^{p-2}u &&\mbox{ in }{\mathbb R}^N, -Δϕ&=K(x)|u|^q&&\mbox{ in }{\mathbb R}^N, \right. $$ where $N\ge 3$, $\hat{I}\pm \in (0,N)$, $p,q>1$ and $K\ge 0$. Using a Pohozaev type identity we first derive conditions in terms of $p,q,N,\hat{I}\pm$ and $K$ for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach.

19 pages