New existence and symmetry results for least energy positive solutions of Schr\"odinger systems with mixed competition and cooperation terms

TL;DR

This paper investigates the existence and symmetry of least energy positive solutions to a cubic Schrödinger system with mixed cooperation and competition terms, providing new theoretical results applicable in bounded and unbounded domains.

Contribution

It introduces novel existence and symmetry results for least energy positive solutions in Schrödinger systems with mixed interaction coefficients, extending previous work to new parameter regimes.

Findings

Established existence of solutions with mixed cooperation and competition terms.

Proved symmetry properties of solutions in various domain settings.

Extended results to purely cooperative cases.

Abstract

In this paper we focus on existence and symmetry properties of solutions to the cubic Schr\"odinger system \[ -\Delta u_i +\lambda_i u_i = \sum_{j=1}^d \beta_{ij} u_j^2 u_i \quad \text{in $\Omega \subset \mathbb{R}^N$},\qquad i=1,\dots d \] where $d\geq 2$, $\lambda_i,\beta_{ii}>0$, $\beta_{ij}=\beta_{ji}\in \mathbb{R}$ for $j\neq i$, $N=2,3$. The underlying domain $\Omega$ is either bounded or the whole space, and $u_i\in H^1_0(\Omega)$ or $u_i\in H^1_{rad}(\mathbb{R}^N)$ respectively. We establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients, as well as in the purely cooperative case.