A coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent: Existence and multiplicity of high energy positive solutions

Abstract

This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent{−Δu+(V1(x)+λ1)u=μ1(|x|−4⁎u2)u+β(|x|−4⁎v2)u,x∈RN,−Δv+(V2(x)+λ2)v=μ2(|x|−4⁎v2)v+β(|x|−4⁎u2)v,x∈RN, where N≥5, λ1, λ2≥0 and λ1+λ2≠0, V1(x),V2(x)∈LN2(RN) are nonnegative functions and μ1, μ2, β are positive constants. Such system arises from mathematical models in Bose-Einstein condensates theory and nonlinear optics. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of high energy positive solutions under the hypothesis β>max⁡{μ1,μ2}.