Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity

TL;DR

This paper investigates the existence and multiplicity of solutions for a doubly nonlocal fractional Laplacian system with critical Hardy-Littlewood-Sobolev nonlinearity, establishing conditions for multiple solutions via variational methods.

Contribution

It introduces a new approach to find multiple solutions for a fractional nonlocal system with critical nonlinearity using fibering maps and Nehari manifold techniques.

Findings

Existence of at least two nontrivial solutions under certain parameters.

Application of fibering map analysis to a fractional nonlocal system.

Identification of parameter ranges for solution multiplicity.

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} (-\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 $\Omega$ is a smooth bounded domain in $\mb R^n$, $n >2s$, $s \in (0,1)$, $(-\Delta)^s$ is the well known fractional Laplacian, $\mu \in (0,n)$, $2^*_\mu = \displaystyle\frac{2n-\mu}{n-2s}$ 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 $(P_{\la,\delta})$ for suitable range of $\la$ and $\delta$.