Least energy positive solutions of critical Schr\"{o}dinger systems with mixed competition and cooperation terms: the higher dimensional case

TL;DR

This paper investigates the existence of least energy positive solutions for critical Schrödinger systems with multiple equations in high dimensions, considering various interaction types and phase separation phenomena, extending known results especially for dimensions four and five.

Contribution

It provides new sufficient conditions for existence of solutions in multi-component Schrödinger systems with mixed cooperation and competition, including phase separation and sign-changing solutions in high dimensions.

Findings

Existence of least energy positive solutions under various coefficient conditions.

Identification of phenomena related to phase separation in multi-component systems.

Proof of existence of least energy sign-changing solutions in dimensions 4 and 5.

Abstract

Let $\Omega\subset \mathbb{R}^{N}$ be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schr\"{o}dinger system with $d\geq 2$ equations \begin{equation*} -\Delta u_{i}+\lambda_{i}u_{i}=|u_{i}|^{p-2}u_{i}\sum_{j = 1}^{d}\beta_{ij}|u_{j}|^{p} \text{ in } \Omega, \quad u_i=0 \text{ on } \partial \Omega, \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 ($\beta_{ii}>0$ for every $i$) in the variational setting $\beta_{ij}=\beta_{ji}$ for every $i\neq j$, dealing with a Br\'ezis-Nirenberg type problem: $-\lambda_{1}(\Omega)<\lambda_{i}<0$, where $\lambda_{1}(\Omega)$ is the first eigenvalue of $(-\Delta,H^1_0(\Omega))$. We provide several sufficient conditions on the coefficients $\beta_{ij}$ that ensure the existence of least energy positive solutions; these include the situations of pure cooperation ($\beta_{ij}> 0$ for every $i\neq j$), pure competition ($\beta_{ij}\leq 0$ for every $i\neq j$) and coexistence of both cooperation and competition coefficients. Some proofs depend heavily on the fact that $1<p<2$, revealing some different phenomena comparing to the special case $N=4$. Our results provide a rather complete picture in the particular situation where the components are divided in two groups. Besides, based on the results about a phase separation phenomena, we prove the existence of least energy sign-changing solution to the Br\'ezis-Nirenberg problem \[ -\Delta u+\lambda u=\mu |u|^{2^*-2}u,\quad u\in H^1_0(\Omega), \] for $\mu>0$, $-\lambda_1(\Omega)<\lambda<0$ for all $N\geq 4$, a result which is new in dimensions $N=4,5$.