High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents

TL;DR

This paper investigates positive solutions for a critical coupled Hartree system with Hardy-Littlewood-Sobolev exponents, classifies solutions in the zero potential case, and proves existence of high energy solutions with small potentials.

Contribution

It classifies positive solutions in the zero potential case and establishes existence of high energy solutions for small nonzero potentials.

Findings

Classified positive solutions using moving sphere method.

Proved uniqueness of positive solutions up to translation and dilation.

Established existence of high energy solutions under small potential conditions.

Abstract

We study the coupled Hartree system $$ \left\{\begin{array}{ll} -\Delta u+ V_1(x)u =\alpha_1\big(|x|^{-4}\ast u^{2}\big)u+\beta \big(|x|^{-4}\ast v^{2}\big)u &\mbox{in}\ \mathbb{R}^N,\\[1mm] -\Delta v+ V_2(x)v =\alpha_2\big(|x|^{-4}\ast v^{2}\big)v +\beta\big(|x|^{-4}\ast u^{2}\big)v &\mbox{in}\ \mathbb{R}^N, \end{array}\right. $$ where $N\geq 5$, $\beta>\max\{\alpha_1,\alpha_2\}\geq\min\{\alpha_1,\alpha_2\}>0$, and $V_1,\,V_2\in L^{N/2}(\mathbb{R}^N)\cap L_{\text{loc}}^{\infty}(\mathbb{R}^N)$ are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ moving sphere arguments in integral form to classify positive solutions and to prove the uniqueness of positive solutions up to translation and dilation, which is of independent interest. Then using the uniqueness property, we establish a nonlocal version of the global compactness lemma and prove the existence of a high energy positive solution for the system assuming that $|V_1|_{L^{N/2}(\mathbb{R}^N)}+|V_2|_{L^{N/2}(\mathbb{R}^N)}>0$ is suitably small.