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

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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth and bounded domain, $n\ge 3$, $a<-{\mathrm{\Lambda }}_1$ where ${\mathrm{\Lambda }}_1$ is the principal eigenvalue of $(-\mathrm{\Delta },H_0^1(\mathrm{\Omega }))$; ${\mu }_j$ and $\beta$ are real constants. Using the positive and non-degenerate solution of the scalar equation $-\mathrm{\Delta }\omega -\omega =-{\omega }^3$, $\omega \in H_0^1(\mathrm{\Omega })$, 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.