On a class of mixed Choquard-Schr\"odinger-Poisson system

TL;DR

This paper investigates a class of mixed Choquard-Schrödinger-Poisson systems, deriving nonexistence conditions and establishing the existence of ground state solutions using variational methods.

Contribution

It provides new conditions for nonexistence and proves the existence of ground states for a complex coupled PDE system.

Findings

Derived nonexistence conditions via Pohozaev identity.

Established existence of ground state solutions.

Analyzed parameter regimes for solution existence.

Abstract

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