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 −ε2Δv + V (x) v = ε 1 α (Iα ∗ F(v))f(v) in RN , where N ≥ 3, α ∈ (0, N), Iα(x) = Aα/|x|N-α is the Riesz potential, F ∈ C1(R, R), F´(s) = f(s) and ε > 0 is a small parameter. We develop a new variational approach and we show the existence of a family of solutions concentrating, as ε → 0, to a local minima of V (x) under general conditions on F(s). Our result is new also for f(s) = |s|p-2s and applicable for p ∈ (N N +α, N N + - α 2 ). Especially, we can give the existence result for locally sublinear case p ∈ (N N +α , 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 ε → 0, where K ⊂ Ω is the set of minima of V (x) in a bounded potential well Ω, that is, m0 ≡ infxεΩ V (x) < infxε∂Ω V (x) and K = {x ∈ Ω; V (x) = m0}.