Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well

We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation -epsilon(2) Delta v + V(x) v = 1/epsilon(alpha) (I-alpha * F(v)) f(v) in R-N, where N >= 3, alpha is an element of (0, N), I-alpha(x) = A(alpha)/vertical bar x vertical bar(N-alpha )is the Riesz potential, F is an element of C-1 (R,R), F' (s) f (s) and epsilon > 0 is a small parameter. We develop a new variational approach and we show the existence of a family of solutions concentrating, as epsilon -> 0, to a local minima of V(x) under general conditions on F(s). Our result is new also for f(s) = vertical bar s vertical bar(p-2)s and applicable for p is an element of (N+alpha/N, N+alpha/N-2). Especially, we can give the existence result for locally sublinear case p is an element of (N+alpha/N, 2), which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen. We also study the multiplicity of positive single-peak solutions and we show the existence of at least cupl(K)+1 solutions concentrating around K as epsilon -> 0, where K subset of Omega is the set of minima of V (x) in a bounded potential well Omega, that is, m(o) inf(x is an element of Omega) V(x) < inf(x is an element of partial derivative Omega) V(x) and K = {x is an element of Omega; V (X) = m(0)}.