Singularly perturbed critical Choquard equations

In this paper we study the semiclassical limit for the singularly perturbed Choquard equation -epsilon(2) Delta u + V(x)u = epsilon(mu-3) (integral(R3) Q(y)G(u(y))/vertical bar x-y vertical bar(mu) dy)Q(x)g(u) in R-3, where 0 < mu < 3, s is a positive parameter, V, Q are two continuous real function on R-3 and G is the primitive of g which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on g, we first establish the existence of ground states for the critical Choquard equation with constant coefficients. Next we establish existence and multiplicity of semi-classical solutions and characterize the concentration behavior by variational methods. (C) 2017 Elsevier Inc. All rights reserved.