Positive least energy solutions for k-coupled critical systems involving fractional Laplacian

In this paper, we study the following $ k $-coupled critical system:

$ \begin{equation*} \left\{ \begin{aligned} (-\Delta)^s u_i +\lambda_iu_i & = \mu_i u_i^{2^*-1}+\sum\limits_{j = 1,j\ne i}^{k} \beta_{ij} u_{i}^{\frac{2^*}{2}-1}u_{j}^{\frac{2^*}{2}} \quad \hbox{in}\;\Omega,\\ u_i & = 0 \quad \hbox{in}\;\; \; \mathbb R^N\backslash\Omega, \quad i = 1,2,\cdots, k. \end{aligned} \right. \end{equation*} $

Here $ (-\Delta)^s $ is the fractional Laplacian operator, $ 0<s<1 $, $ 2^{*} = \frac{2N}{N-2s} $ is a fractional Sobolev critical exponent, $ N>2s $, $ - \lambda_s( \Omega)< \lambda_i<0, \mu_i>0 $, $ \beta_{ij} = \beta_{ji}\ne 0 $ and $ \Omega\subset {\mathbb R}^N $ is a smooth bounded domain, where $ \lambda_s( \Omega) $ is the first eigenvalue of $ (-\Delta)^{s} $ with the homogeneous Dirichlet boundary datum. We characterize the positive least energy solution of the $ k $-coupled fractional critical system for the purely cooperative case $ \beta_{ij}>0 $ with $ N> 4s $. We shall introduce the idea of induction to prove our results. We point out that the key idea is to give a more accurate upper bound of the least energy. It's interesting to see that the least energy of the $ k $-coupled system decreases as $ k $ grows. Moreover, we establish the existence of positive least energy solution of the limit system in $ \mathbb R^N $, as well as classification results. Meanwhile, we also construct a positive solution for a more general system involving subcritical items. Besides, we investigated in the asymptotic behaviour of the positive least energy solutions of the critical system. We point out that the results of the fractional critical systems have some coincidences with those of the critical Schrödinger systems.