Blow-up of ground states of fractional Choquard equations

Abstract

We investigate the blow-up behavior of ground states of the fractional Choquard equation (−Δ)su+u=(Kα∗|u|pϵ)|u|pϵ−2uin RN,with s∈(0,1), N>4s, Kα Riesz potential of order α∈(0,N), as the exponent pϵ approaches the upper critical growth regime in Hardy–Littlewood–Sobolev’s inequality. We prove that the ground state uϵ blows up in the sense that ‖uϵ‖L∞(RN)=oϵ−N−2s4s as ϵ→0+.