Positive least energy solutions for k-coupled Schrödinger system with critical exponent: the higher dimension and cooperative case

Abstract

In this paper, we study the following k-coupled nonlinear Schrödinger system with Sobolev critical exponent: $$\begin{aligned} \left\{ \begin{aligned} -\Delta u_i&+\lambda _iu_i =\mu _i u_i^{2^*-1}+\sum _{j=1,j\ne i}^{k} \beta _{ij} u_{i}^{\frac{2^*}{2}-1}u_{j}^{\frac{2^*}{2}} \quad \hbox {in}\;\Omega ,\\ u_i&>0 \quad \hbox {in}\; \Omega \quad \hbox {and}\quad u_i=0 \quad \hbox {on}\;\partial \Omega , \quad i=1,2,\dots , k. \end{aligned} \right. \end{aligned}$$Here \(\Omega \subset {{\mathbb {R}}}^N \) is a smooth bounded domain, \(2^{*}=\frac{2N}{N-2}\) is the Sobolev critical exponent, \(-\lambda _1(\Omega )<\lambda _i<0, \mu _i>0\) and \( \beta _{ij}=\beta _{ji}\ne 0\), where \(\lambda _1(\Omega )\) is the first eigenvalue of \(-\Delta \) 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 delicate, 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 is 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.