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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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