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 ∗ (|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.