On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents

Abstract

We consider the following nonlinear Choquard equation with Dirichlet boundary condition−Δu=(∫Ω|u|2μ⁎|x−y|μdy)|u|2μ⁎−2u+λf(u)inΩ, where Ω is a smooth bounded domain of RN, λ>0, N≥3, 0<μ<N and 2μ⁎ is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Under suitable assumptions on different types of nonlinearities f(u), we are able to prove some existence and multiplicity results for the equation by variational methods.