Infinitely many solutions for The Brézis-Nirenberg problem with nonlinear Choquard equations

Abstract

In this paper, we consider the following Brézis-Nirenberg problem with the Choquard equations:{−Δu=∫Ω|u(y)|2α⁎|x−y|N−αdy|u|2α⁎−2u+λuin Ω,u=0on ∂Ω, where α∈(N−4,N), Ω is a bounded smooth domain in RN(N≥7) and 2α⁎=N+αN−2 is the Hardy-Littlewood-Sobolev critical exponent. We show that, for each λ>0, this problem has infinitely many solutions by using truncation method.