Existence and concentration of ground states for a Choquard equation with competing potentials

Abstract

In this paper, we are concerned with the following Choquard equation in R3 that−ϵ2Δu+V(x)u=ϵμ−3[(∫R3P(y)|u(y)|p|x−y|μ)P(x)|u|p−2u+(∫R3Q(y)|u(y)|q|x−y|μ)Q(x)|u|q−2u], where ϵ>0 is a parameter, 0<μ<3, 6−μ3<q<p<6−μ, the functions V and P are positive and Q may be sign-changing. Via variational methods, we establish the existence of ground states for small ϵ, and investigate the concentration behavior of ground states and show that they concentrate at a global minimum point of the least energy function as ϵ→0.