Bifurcations for a Coupled Schrödinger System with Multiple Components

Abstract

In this paper, we study local bifurcations of an indefinite elliptic system with multiple components: \begin{equation*} \left\{\begin{array}{ll} -Δu_j + au_j = μ_ju_j^3+β\sum_{k\ne j}u_k^2u_j, u_j>0\ \ \hbox{in}\ Ω, u_j=0 \ \ \hbox{on}\ \partialΩ,\ j=1,\dots,n. \end{array} \right. \end{equation*} Here $\Omega \subset {\mathbb{R}}^N$ is a smooth and bounded domain, $n\ge 3$, $a<-{\Lambda }_1$ 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.

16 pages, 2 figures

References in corpus (1)

• Bifurcation in a multi-component system of nonlinear Schrödinger equations