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

作者：Chen, Mengyao^*; Guo, Lun^*; Li, Qi^*

来源：Journal of Differential Equations, 2024, 385: 1-56.

DOI：10.1016/j.jde.2023.12.001

• Hartree system
• Hardy-Littlewood-Sobolev critical exponent
• Lack of compactness
• Positive solution

This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent {-Delta u + (V-1(x) + lambda(1))u = mu(1) (|x|(-4) & lowast; u(2))u + beta(|x|(-4) & lowast; v(2))u, x is an element of R-N, {-Delta v + (V-2(x) + lambda(2))v = mu(2)(|x|(-4) & lowast; v(2))v + beta(|x|(-4) & lowast; u(2))v, x is an element of R-N, where N >= 5, lambda(1), lambda(2) >= 0 and lambda 1 + lambda 2 not equal 0, V-1(x), V-2(x) is an element of L (N/2) (R-N) are nonnegative functions and mu(1), mu(2), beta 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 beta > max{mu(1), mu(2)}.