Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity

This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left{ \begin{split} (-\Delta)^su &= \lambda |u|^{q-2}u + \left(\int_{\Omega}\frac{|v(y)|^{2^_\mu}}{|x-y|^\mu}~$\mathrm{d}$y\right) |u|^{2^\mu-2}u; $\text{in}$; \Omega (-\Delta)^sv &= \delta |v|^{q-2}v + \left(\int{\Om}\frac{|u(y)|^{2^_\mu}}{|x-y|^\mu}~$\mathrm{d}$y \right) |v|^{2^_\mu-2}v ; $\text{in}$; \Omega u &=v=0; $\text{in}$; \mb R^n\setminus\Omega, \end{split} \right. \end{equation*} where $\mathrm{\Omega }$ is a smooth bounded domain in $\text{mb}{R}^n$, $n

2s$,$s \in (0,1)$,$(-\Delta)^s$isthewellknownfractionalLaplacian,$\mu \in (0,n)$,$2^*_\mu = \displaystyle\frac{2n-\mu}{n-2s}$ is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality, $1<q<2$ and $\lambda ,\delta >0$ are real parameters. We study the fibering maps corresponding to the functional associated with $(P_{\lambda ,\delta })$ and show that minimization over suitable subsets of Nehari manifold renders the existence of atleast two non trivial solutions of $(P_{\text{la},\delta })$ for suitable range of $\text{la}$ and $\delta$.