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

Silvia Cingolani, Kazunaga Tanaka

• We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation −ε2Δv+V(x)v=1εα(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+αN,N+αN−2). Especially, we can give the existence result for locally sublinear case p∈(N+αN,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≡infx∈ΩV(x)