Least energy positive solutions of critical Schrödinger systems with mixed competition and cooperation terms: the higher dimensional case

Abstract

Let $Ω\subset \mathbb{R}^{N}$ be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schrödinger system with $d\geq 2$ equations \begin{equation*} -Δu_{i}+λ_{i}u_{i}=|u_{i}|^{p-2}u_{i}\sum_{j = 1}^{d}β_{ij}|u_{j}|^{p} \text{ in } Ω, \quad u_i=0 \text{ on } \partial Ω, \qquad i=1,...,d, \end{equation*} in the case of a critical exponent $2p=2^*=\frac{2N}{N-2}$ in high dimensions $N\geq 5$. We treat the focusing case ($β_{ii}>0$ for every $i$) in the variational setting $β_{ij}=β_{ji}$ for every $i\neq j$, dealing with a Brézis-Nirenberg type problem: $-λ_{1}(Ω)<λ_{i}<0$, where $λ_{1}(Ω)$ is the first eigenvalue of $(-Δ,H^1_0(Ω))$. We provide several sufficient conditions on the coefficients $β_{ij}$ that ensure the existence of least energy positive solutions; these include the situations of pure cooperation ($β_{ij}> 0$ for every $i\neq j$), pure competition ($β_{ij}\leq 0$ for every $i\neq j$) and coexistence of both cooperation and competition coefficients. Some proofs depend heavily on the fact that $1 0$, $-λ_1(Ω)<λ<0$ for all $N\geq 4$, a result which is new in dimensions $N=4,5$.