Bound states of fractional Choquard equations with Hardy-Littlewood-Sobolev critical exponent

Abstract

We deal with the following fractional Choquard equation(−Δ)su+V(x)u=(Iμ⁎|u|2μ,s⁎)|u|2μ,s⁎−2u,x∈RN, where Iμ(x) is the Riesz potential, s∈(0,1), 2s<N≠4s, 0<μ<min⁡{N,4s} and 2μ,s⁎=2N−μN−2s is the fractional critical Hardy-Littlewood-Sobolev exponent. By combining variational methods and the Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this equation when V(x) is a positive potential bounded from below. The results obtained in this paper extend and improve some recent works in the case where the coefficient V(x) vanishes at infinity.