Ground state of a magnetic nonlinear Choquard equation

Abstract

We consider the stationary magnetic nonlinear Choquard equation −(∇+iA(x))2u+V(x)u=(1|x|α∗F(|u|))f(|u|)|u|u,where A:RN→RN is a vector potential, V is a scalar potential, f:R→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.