Existence results for non-local elliptic systems with Hardy-Littlewood-Sobolev critical nonlinearities
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} (-Δ)^s u & = au+bv+\frac{2p}{p+q}\int_Ω\frac{|v(y)|^q}{|x-y|^μ}dy|u|^{p-2}u+2ξ_1\int_Ω\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy|u|^{2^*_μ-2}u,&& \text{in } Ω;\\ (-Δ)^s v & = bu+cv+\frac{2q}{p+q}\int_Ω\frac{|u(y)|^p}{|x-y|^μ}dy|v|^{q-2}v+2ξ_2\int_Ω\frac{|v(y)|^{2^*_μ}}{|x-y|^μ}dy|v|^{2^*_μ-2}v,&& \text{in } Ω;\\ u &=v=0,\text{ in } \R^N\setminusΩ, \end{aligned}\right. \end{equation*} where $Ω$ is a smooth bounded domain in $\R^N$, $N>2s$, $s\in(0,1)$, $ξ_1,ξ_2\geq 0$, $(-Δ)^s$ is the well known fractional Laplacian, $μ\in(0,N)$, $1<p,q\leq 2^*_μ$ where $2^*_μ=\frac{2N-μ}{N-2s}$ is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different parameters $p, q, ξ_1,$ and $ ξ_2$, we are able to prove some existence and multiplicity results for the above equation by variational methods.