Positive least energy solutions for $k$-coupled Schr\"odinger system with critical exponent: the higher dimension and cooperative case

In this paper, we study the following $k$-coupled nonlinear Schr"odinger system with Sobolev critical exponent: \begin{equation*} \left{ \begin{aligned} -\Delta u_i & +\lambda_iu_i =\mu_i u_i^{2^-1}+\sum_{j=1,j\ne i}^{k} \beta_{ij} u_{i}^{\frac{2^}{2}-1}u_{j}^{\frac{2^}{2}} \quad \hbox{in};\Omega,\newline u_i&>0 \quad \hbox{in}; \Omega \quad \hbox{and}\quad u_i=0 \quad \hbox{on};\partial\Omega, \quad i=1,2,\cdots, k. \end{aligned} \right. \end{equation} Here $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $2^{\ast }=\frac{2N}{N-2}$ is the Sobolev critical exponent, $-{\lambda }_1(\mathrm{\Omega })<{\lambda }_i<0,{\mu }_i>0$ and ${\beta }_{ij}={\beta }_{ji}\mathrm{\ne }0$, where ${\lambda }_1(\mathrm{\Omega })$ is the first eigenvalue of $-\mathrm{\Delta }$ with the Dirichlet boundary condition. We characterize the positive least energy solution of the $k$-coupled system for the purely cooperative case ${\beta }_{ij}>0$, in higher dimension $N\ge 5$. Since the $k$-coupled case is much more delicated, we shall introduce the idea of induction. 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.