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

TL;DR

This paper investigates positive least energy solutions for a $k$-coupled nonlinear Schrödinger system with critical exponent in higher dimensions, revealing energy bounds, existence, and classification results, especially in the cooperative case.

Contribution

It introduces an induction method to analyze the $k$-coupled system, providing new bounds and demonstrating how least energy decreases with increasing $k$ in higher dimensions.

Findings

Least energy solutions characterized for the cooperative case in high dimensions.

The least energy decreases as the number of coupled equations increases.

Existence and classification of solutions in the limit system in \\mathbb{R}^N.

Abstract

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 $\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 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.