Solutions for a class of quasilinear Choquard equations with Hardy–Littlewood–Sobolev critical nonlinearity

Abstract

In this article, we consider the quasilinear Choquard equation with critical nonlinearity in RN: −ε2Δu+V(x)u−ε2uΔu2=∫RN|u|22μ∗|x−y|μdy|u|22μ∗−2u+h(x,u)u,where 0<μ<N, N≥3, 2μ∗=(2N−μ)∕(N−2) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, ε is a real parameter. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Under suitable assumptions on V and h, we investigate the existence and multiplicity of solutions for the above problem by using the mountain pass theorem and index theory. In order to overcome the lack of compactness, we apply the concentration-compactness principle.