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Existence and bifurcation of solutions for a double coupled system of Schrödinger equations. (English) Zbl 1326.35133
Summary: Consider the following system of double coupled Schrödinger equations arising from Bose-Einstein condensates etc., \[ \begin{cases} -\Delta u + u = \mu _1 u^3 + \beta uv^2 - \kappa v,\\ -\Delta v+v=\mu _2 v^3 + \beta u^2 v - \kappa u,\\ u \neq\, 0,v \neq 0,\text{ and }u,v\in H^1 (\mathbb R^N ),\end{cases} \] where \(\mu_1\), \(\mu_2\) are positive and fixed; \(\kappa\) and \(\beta\) are linear and nonlinear coupling parameters respectively. We first use critical point theory and Liouville type theorem to prove some existence and nonexistence results on the positive solutions of this system. Then using the positive and non-degenerate solution to the scalar equation \(-\Delta\omega +\omega =\omega^3\), \(\omega\in H_r^1(\mathbb R^N)\), we construct a synchronized solution branch to prove that for \(\beta\) in certain range and fixed, there exist a series of bifurcations in product space \(\mathbb R \times H_r^1(\mathbb R^N)\times H_r^1(\mathbb R^N)\) with parameter \(\kappa\).

MSC:
35J50 Variational methods for elliptic systems
35B32 Bifurcations in context of PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35B09 Positive solutions to PDEs
35J47 Second-order elliptic systems
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