MASS MINIMIZERS AND CONCENTRATION FOR NONLINEAR CHOQUARD EQUATIONS IN RN

In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: E(u) = 1/2 integral(RN) |del u|(2) + 1/2 integral(RN) V(x)|u|(2) - 1/2p integral(RN) (I-alpha * |u|(p))|u|(p) on (S) over tilde (c) = {u is an element of H-1(R-N) | integral(RN) V(x)|u|(2) < +infinity, |u|(2) = c, c > 0} where N >= 1, alpha is an element of (0, N), (N + alpha)/N <= p < (N + alpha)/(N - 2)(+) and I-alpha: R-N -> R is the Riesz potential. We present sharp existence results for E(u) constrained on <(S)over tilde>(c) when V(x) equivalent to 0 for all (N + alpha)/N <= p < (N + alpha)/(N - 2)(+). For the mass critical case p = (N + alpha + 2)/N, we show wthat if 0 <= V is an element of L-loc(infinity)(R-N) and lim(|x|->+infinity) V(x) = +infinity, then mass minimizers exist only if 0 < c < c* = |Q|(2) and concentrate at the flattest minimum of V as c approaches c* from below, where Q is a groundstate solution of -Delta u + u = (I alpha * |mu|((N+alpha+2)/N))|u|((N+alpha+2)/N-2)u in R-N.