Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity

Abstract

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} (-Δ)^su &= λ|u|^{q-2}u + \left(\int_Ω\frac{|v(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y\right) |u|^{2^*_μ-2}u\; \text{in}\; Ω (-Δ)^sv &= δ|v|^{q-2}v + \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y \right) |v|^{2^*_μ-2}v \; \text{in}\; Ω u &=v=0\; \text{in}\; \mb R^n\setminusΩ, \end{split} \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\mb R^n$, $n >2s$, $s \in (0,1)$, $(-Δ)^s$ is the well known fractional Laplacian, $μ\in (0,n)$, $2^*_μ= \displaystyle\frac{2n-μ}{n-2s}$ is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality, $1 0$ are real parameters. We study the fibering maps corresponding to the functional associated with $(P_{λ,δ})$ and show that minimization over suitable subsets of Nehari manifold renders the existence of atleast two non trivial solutions of $(P_{\la,δ})$ for suitable range of $\la$ and $δ$.