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Bound states for a coupled Schrödinger system. (English) Zbl 1153.35390
Summary: We consider the existence of bound states for the coupled elliptic system
\[ \begin{aligned}\Delta u_1 - \lambda _1 u_1 + \mu_1 u_1^3 + \beta u_2^2 u_1 = 0\quad \text{in }{\mathbb{R}}^n, \\ \Delta u_2 - \lambda _2 u_2 + \mu_2 u_2^3 + \beta u_1^2 u_2 = 0\quad \text{in }{\mathbb{R}}^n, \\ u_1 > 0,\quad u_2 > 0,\quad u_1 ,u_2 \in {\mathbb{H}}^1({\mathbb{R}}^n), \end{aligned} \] where \(n \leq 3\). Using the fixed point index in cones we prove the existence of a five-dimensional continuum \({\mathcal{C}}\subset {\mathbb{R}}_+^5 \times {\mathbb{H}}^1 ({\mathbb{R}}^n)\times {\mathbb{H}}^1 ({\mathbb{R}}^n)\) of solutions \((\lambda _{1}, \lambda _{2}, \mu _{1}, \mu _{2}, \beta, u _{1}, u _{2})\) bifurcating from the set of semipositive solutions (where \(u_{1} = 0\) or \(u_{2} = 0\)) and investigate the parameter range covered by \({\mathcal{C}}\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35B32 Bifurcations in context of PDEs
35J50 Variational methods for elliptic systems
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
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