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

TL;DR

This paper investigates the existence and multiplicity of high energy positive solutions for a coupled Hartree system with critical Hardy-Littlewood-Sobolev exponent, relevant to Bose-Einstein condensates and nonlinear optics, using variational and degree theory methods.

Contribution

It establishes new results on solutions' existence and multiplicity for a critical coupled Hartree system with nonnegative potentials, under specific parameter conditions.

Findings

Proved existence of high energy positive solutions.

Established multiplicity results under certain parameter regimes.

Applied variational methods combined with degree theory.

Abstract

This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent \begin{equation*} \begin{cases} -\Delta u+(V_1(x)+\lambda_1)u=\mu_1(|x|^{-4}*u^{2})u+\beta (|x|^{-4}*v^{2})u, \ \ &x\in R^N, -\Delta v+(V_2(x)+\lambda_2)v=\mu_2(|x|^{-4}*v^{2})v+\beta (|x|^{-4}*u^{2})v, \ \ &x\in R^N, \end{cases} \end{equation*} where $N\geq 5$, $\lambda_1$, $\lambda_2\geq 0$ with $\lambda_1+\lambda_2\neq 0$, $V_1(x), V_{2}(x)\in L^{\frac{N}{2}}(R^N)$ 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 $\beta>\max\{\mu_1,\mu_2\}$