Existence of positive solutions for fractional Laplacian systems with critical growth

Abstract

In this article, we show the existence of positive solution to the nonlocal system $$\displaylines{ (-\Delta)^s u +a(x)u=\frac{1}{2_s^*} H_u(u,v) \quad \hbox{in }\mathbb{R}^N,\cr (-\Delta)^s v +b(x)v=\frac{1}{2_s^*}H_v(u,v) \quad \hbox{in } \mathbb{R}^N,\cr u,v>0 \quad \text{in } \mathbb{R}^N,\cr u,v\in \mathcal{D}^{s,2 }(\mathbb{R}^N). }$$ We also prove a global compactness result for the associated energy functionalsimilar to that due to Struwe in [26]. The basic tools are some information from a limit systemwith \(a(x) = b(x) = 0\), a variant of the Lion's principle of concentration and compactness for fractional systems, and Brouwer degree theory.