Multiple Positive Solutions for a Fractional Laplacian System with Critical Nonlinearities

Abstract

In this paper, we study the following nonlinear fractional Laplacian system with critical exponent $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda |u|^{p-2}u+\frac{2\alpha }{\alpha +\beta }|u|^{\alpha -2}u|v|^{\beta }, &{}\quad \hbox {in} \;\ \Omega ,\\ (-\Delta )^{s}v=\mu |v|^{p-2}v+\frac{2\beta }{\alpha +\beta }|u|^{\alpha }|v|^{\beta -2}v, &{} \quad \hbox {in} \;\ \Omega ,\\ u=v=0, &{} \quad \hbox {in} \;\ {\mathbb {R}}^{N}\backslash \Omega , \end{array}\right. \end{aligned}$$ where \(\Omega \subset {\mathbb {R}}^{N}\) is a bounded domain with smooth boundary, \(0 1\) satisfy \(\alpha +\beta =2_{s}^{*}, 2_{s}^{*}=\frac{2N}{N-2s}\) is the critical Sobolev exponent, and \(N>4s, \lambda , \mu >0\) are parameters. Using the \({\mathcal {N}}\) ehari manifold, fibering maps and the Lusternik–Schnirelmann category, we prove that the problem has at least \(\hbox {cat}(\Omega )+1\) distinct positive solutions, where \(\hbox {cat} (\Omega )\) denotes the Lusternik–Schnirelmann category of \(\Omega \) in itself.