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Bifurcations for a coupled Schrödinger system with multiple components. (English) Zbl 1330.35134
Summary: In this paper, we study local bifurcations of an indefinite elliptic system with multiple components: $\begin{cases}-\Delta{u_{j}} + au_{j} = \mu_{j}u_{j}^3 + \beta \sum_{k \neq j}u_{k}^2u_{j},\\ u_{j} > 0 {\text{ in}}\,\Omega, u_{j} = 0 {\text{ on}}\, \partial \Omega, j = 1,\dots,n. \end{cases}$ Here $${\Omega \subset\mathbb{R}^N}$$ is a smooth and bounded domain, $${n \geq 3, a < - \Lambda_1 {\text{ where}}\, \Lambda_1}$$ is the principal eigenvalue of $$(-\Delta, H_{0}^1(\Omega))$$; $$\mu_{j}$$ and $$\beta$$ are real constants. Using the positive and nondegenerate 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.

##### MSC:
 35J57 Boundary value problems for second-order elliptic systems 35B09 Positive solutions to PDEs 35B32 Bifurcations in context of PDEs 35J10 Schrödinger operator, Schrödinger equation 35J47 Second-order elliptic systems 35J61 Semilinear elliptic equations 58C40 Spectral theory; eigenvalue problems on manifolds
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