Standing waves with a critical frequency for nonlinear Choquard equations

We study the nonlocal Choquard equation -epsilon(2) Delta u(epsilon) + Vu(epsilon) = (I-alpha *|u(epsilon)|(p))|u(epsilon)|(p-2)u(epsilon) in R-N where N >= 1,I-alpha is the Riesz potential of order alpha is an element of (0, N) and epsilon > 0 is a parameter. When the nonnegative potential V is an element of C(R-N) achieves 0 with a homogeneous behavior or on the closure of an open set but remains bounded away from 0 at infinity, we show the existence of groundstate solutions for small epsilon > 0 and exhibit the concentration behavior as epsilon -> 0. (C) 2017 Elsevier Ltd. All rights reserved.