Semiclassical states for critical Choquard equations

Abstract

In this paper, by constructing an appropriate Byeon-Wang type penalization function, we consider the critical Choquard equation−ϵ2Δu+V(x)u=1ϵα(Iα⁎|u|N+αN−2)|u|N+αN−2−2u+λg(u)inRN, where N≥3, (N−4)+<α<N, ϵ,λ>0 are parameters, and Iα is the Riesz potential. We obtained the localized semiclassical states concentrating at local minimizers of potential. It is the first time to study Choquard equations using this method as far as we know.