Ground state of a magnetic nonlinear Choquard equation

Abstract

We consider the stationary magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u=\bigg(\frac{1}{|x|^α}*F(|u|)\bigg)\frac{f(|u|)}{|u|}{u},\] where $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is a vector potential, $V$ is a scalar potential, $f:\mathbb{R}\to \mathbb{R}$ and $F$ is the primitive of $f$. Under mild hypotheses, we prove the existence of a ground state solution for this problem. We also prove a simple multiplicity result by applying Ljusternik-Schnirelmann methods.

11 pages