Bifurcations for a Coupled Schr\"odinger System with Multiple Components

TL;DR

This paper investigates local and global bifurcations in a coupled multi-component Schrödinger system with indefinite elliptic operators, revealing new solution branches and bifurcation phenomena in bounded domains.

Contribution

It introduces a novel analysis of bifurcations for a multi-component Schrödinger system with indefinite elliptic operators, constructing solution branches and identifying bifurcation points.

Findings

Existence of a synchronized solution branch $\\mathcal{T}_\omega$.

Identification of local bifurcations from the synchronized branch.

Discovery of global bifurcation branches of partially synchronized solutions.

Abstract

In this paper, we study local bifurcations of an indefinite elliptic system with multiple components: \begin{equation*} \left\{\begin{array}{ll} -\Delta u_j + au_j = \mu_ju_j^3+\beta\sum_{k\ne j}u_k^2u_j, u_j>0\ \ \hbox{in}\ \Omega, u_j=0 \ \ \hbox{on}\ \partial\Omega,\ j=1,\dots,n. \end{array} \right. \end{equation*} Here $\Omega\subset{\mathbb{R}}^N$ is a smooth and bounded domain, $n\ge3$, $a<-\Lambda_1$ where $\Lambda_1$ is the principal eigenvalue of $(-\Delta, H_0^1(\Omega))$; $\mu_j$ and $\beta$ are real constants. Using the positive and non-degenerate solution of the scalar equation $-\Delta\omega-\omega=-\omega^3$, $\omega\in H_0^1(\Omega)$, we construct a synchronized solution branch $\mathcal{T}_\omega$. Then we find a sequence of local bifurcations with respect to $\mathcal{T}_\omega$, and we find global bifurcation branches of partially synchronized solutions.