Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity

Abstract

We study the following nonlinear Choquard equation: -Δ u + V(x)u =(1/|x|μ ∗ F(u))f(u) in RN , where 0 < μ < N, N ≥ 3, V is a continuous real function and F is the primitive function of f . Under some suitable assumptions on the potential V, which include the case V(∞ ) = 0, that is, V(x) → 0 as |x| → +∞ we prove the existence of a nontrivial solution for the above equation by the penalization method.