Bifurcations for a Coupled Schrödinger 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} -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 $Ω\subset{\mathbb{R}}^N$ is a smooth and bounded domain, $n\ge3$, $a<-Λ_1$ where $Λ_1$ is the principal eigenvalue of $(-Δ, H_0^1(Ω))$; $μ_j$ and $β$ are real constants. Using the positive and non-degenerate solution of the scalar equation $-Δω-ω=-ω^3$, $ω\in H_0^1(Ω)$, we construct a synchronized solution branch $\mathcal{T}_ω$. Then we find a sequence of local bifurcations with respect to $\mathcal{T}_ω$, and we find global bifurcation branches of partially synchronized solutions.

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