Semiclassical states for fractional Schrödinger equations with critical growth

Abstract

In this paper, we consider the following fractional nonlinear Schrödinger equations ε2s(−Δ)su+V(x)u=P(x)g(u)+Q(x)|u|2s∗−2u,x∈RN and prove the existence and concentration of positive solutions under suitable assumptions on the potentials V(x),P(x) and Q(x). We show that the semiclassical solutions uε with maximum points xε concentrating at a special set SP characterized by V(x),P(x) and Q(x). Moreover, for any sequence xε→x0∈SP, vε(x):=uε(εx+xε) convergence strongly in Hs(RN) to a ground state solution v of (−Δ)sv+V(x0)v=P(x0)g(v)+Q(x0)|v|2s∗−2v,x∈RN.