Least energy positive soultions for $d$-coupled Schrödinger systems with critical exponent in dimension three

Abstract

In the present paper, we consider the coupled Schrödinger systems with critical exponent: \begin{equation*} \begin{cases} -Δu_i+λ_{i}u_i=\sum\limits_{j=1}^{d} β_{ij}|u_j|^{3}|u_i|u_i \quad ~\text{ in } Ω,\\ u_i \in H_0^1(Ω) ,\quad i= 1,2,...,d. \end{cases} \end{equation*} Here, $\Omega \subset {\mathbb{R}}^3$ is a smooth bounded domain, $d\ge 2$, ${\beta }_{ii}>0$ for every $i$, and ${\beta }_{ij}={\beta }_{ji}$ for $i\ne j$. We study a Brézis-Nirenberg type problem: $-{\lambda }_1(\Omega )<{\lambda }_1,\cdots ,{\lambda }_d<-{\lambda }^{\ast }(\Omega )$, where ${\lambda }_1(\Omega )$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary conditions and ${\lambda }^{\ast }(\Omega )\in (0,{\lambda }_1(\Omega ))$. We acquire the existence of least energy positive solutions to this system for weakly cooperative case (${\beta }_{ij}>0$ small) and for purely competitive case (${\beta }_{ij}\le 0$) by variational arguments. The proof is performed by mathematical induction on the number of equations, and requires more refined energy estimates for this system. Besides, we present a new nonexistence result, revealing some different phenomena comparing with the higher-dimensional case $N\ge 5$. It seems that this is the first paper to give a rather complete picture for the existence of least energy positive solutions to critical Schrödinger system in dimension three.