Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 79, pp. 1-42.

Title: Existence of positive solutions for fractional Laplacian systems with critical growth
       
Authors: Jeziel N. Correia (Univ. Federal do Pare, Salinepolis, Brazil}
         Claudionei P. Oliveira (Univ. Federal do Sul e Sudeste do Pare, Marabe, Brazil)
         
Abstract:
In this article, we show the existence of positive solution to the nonlocal system
$$\displaylines{
(-\Delta)^s u +a(x)u=\frac{1}{2_s^*} H_u(u,v) \quad \text{in }\mathbb{R}^N,\cr
(-\Delta)^s v +b(x)v=\frac{1}{2_s^*}H_v(u,v) \quad \text{in }  \mathbb{R}^N,\cr
u,v>0 \quad \text{in } \mathbb{R}^N,\\
u,v\in \mathcal{D}^{s,2}(\mathbb{R}^N).
}$$
We also prove a global compactness result for the associated energy functional
similar to that due to Struwe in [26].
The basic tools are some information from a limit system
with a(x) = b(x) = 0, a variant of the Lion's principle of concentration
and compactness for fractional systems, and Brouwer degree theory. 

Submitted  March 29, 2022. Published November 22, 2022.
Math Subject Classifications: 35J20, 35J47, 35J50, 35J91.
Key Words: Fractional Laplacian; concentration-compactness; critical nonlinearity global compactness.