Concentration of solutions for a fractional relativistic Schrödinger–Choquard equation with critical growth

Abstract

This paper is devoted to the following fractional relativistic Schrödinger–Choquard equation with critical growth: (−ɛ2Δ+m2)su+V(x)u=f(u)+ɛ−αIα∗|u|2α∗|u|2α∗−2u,x∈RN,u∈Hs(RN),u>0,x∈RN,where (−Δ+m2)s is the fractional relativistic Schrödinger operator, s∈(0,1), N>2s, m>0, Iα is a Riesz potential of order α∈(0,N),2α∗=N+αN−2s is the upper critical exponent due to Hardy–Littlewood–Sobolev inequality, V:RN→R is a continuous potential and f:R→R is a super-linear continuous nonlinearity with subcritical growth at infinity. Under the suitable assumptions on the potential V, we construct a family of positive solutions uɛ∈Hs(RN), with exponential decay, which concentrates around a local minimum of V as ɛ→0.