Existence results for non-local elliptic systems with Hardy-Littlewood-Sobolev critical nonlinearities

TL;DR

This paper establishes existence and multiplicity of solutions for a nonlocal elliptic system involving fractional Laplacian and Hardy-Littlewood-Sobolev critical nonlinearities using variational methods.

Contribution

It provides new existence and multiplicity results for a fractional elliptic system with critical nonlinearities, extending previous work to more general nonlocal problems.

Findings

Proved existence of solutions under certain parameter conditions.

Established multiple solutions using variational techniques.

Extended results to systems with Hardy-Littlewood-Sobolev critical nonlinearities.

Abstract

In this article, we study the following nonlinear doubly nonlocal problem involving the fractional Laplacian in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{\begin{aligned} (-\Delta)^s u & = au+bv+\frac{2p}{p+q}\int_{\Omega}\frac{|v(y)|^q}{|x-y|^\mu}dy|u|^{p-2}u+2\xi_1\int_{\Omega}\frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu}dy|u|^{2^*_\mu-2}u,&& \text{in } \Omega;\\ (-\Delta)^s v & = bu+cv+\frac{2q}{p+q}\int_{\Omega}\frac{|u(y)|^p}{|x-y|^\mu}dy|v|^{q-2}v+2\xi_2\int_{\Omega}\frac{|v(y)|^{2^*_\mu}}{|x-y|^\mu}dy|v|^{2^*_\mu-2}v,&& \text{in } \Omega;\\ u &=v=0,\text{ in } \R^N\setminus\Omega, \end{aligned}\right. \end{equation*} where $\Omega$ is a smooth bounded domain in $\R^N$, $N>2s$, $s\in(0,1)$, $\xi_1,\xi_2\geq 0$, $(-\Delta)^s$ is the well known fractional Laplacian, $\mu\in(0,N)$, $1<p,q\leq 2^*_\mu$ where $2^*_\mu=\frac{2N-\mu}{N-2s}$ is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different parameters $p, q, \xi_1,$ and $ \xi_2$, we are able to prove some existence and multiplicity results for the above equation by variational methods.