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

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, $Ω\subset \mathbb{R}^{3}$ is a smooth bounded domain, $d \geq 2$, $β_{ii}>0$ for every $i$, and $β_{ij}=β_{ji}$ for $i \neq j$. We study a Brézis-Nirenberg type problem: $-λ_{1}(Ω)<λ_{1},\cdots,λ_{d}<-λ^*(Ω)$, where $λ_{1}(Ω)$ is the first eigenvalue of $-Δ$ with Dirichlet boundary conditions and $λ^*(Ω)\in (0, λ_1(Ω))$. We acquire the existence of least energy positive solutions to this system for weakly cooperative case ($β_{ij}>0$ small) and for purely competitive case ($β_{ij}\leq 0$) 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 $N\geq 5$. 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.

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