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Existence and bifurcation of solutions for a double coupled system of Schrodinger equations
- Publication Year :
- 2015
-
Abstract
- Consider the following system of double coupled Schr\"odinger equations arising from Bose-Einstein condensates etc., \begin{equation*} \left\{\begin{array}{l} -\Delta u + u =\mu_1 u^3 + \beta uv^2- \kappa v, -\Delta v + v =\mu_2 v^3 + \beta u^2v- \kappa u, u\neq0, v\neq0\ \hbox{and}\ u, v\in H^1(\R^N), \end{array} \right. \end{equation*}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 of the scalar equation $-\Delta\omega+\omega=\omega^3$, $\omega\in H_r^1(\R^N)$, 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 $\R\times H^1_r(\R^N)\times H^1_r(\R^N)$ with parameter $\kappa$.<br />Comment: Science China Mathematics,2015
- Subjects :
- Mathematics - Analysis of PDEs
35B32, 35B38, 35J50, 58C40, 58E07
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1504.06923
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s11425-000-0000-0