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Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger systems. (English) Zbl 1419.35181
Summary: In this paper, we consider the following coupled elliptic system \[\begin{cases} -\Delta u+\lambda_1 u= \mu_1 u^3 + \beta uv^2 - \gamma v \quad & \text{in }\mathbb{R}^N, \\ -\Delta v+\lambda_2 v= \mu_2v^3 +\beta vu^2 - \gamma u & \text{in }\mathbb{R}^N, \\ u(x),v(x) \rightarrow 0 \text{ as }|x|\rightarrow +\infty. \end{cases}\] Under symmetric assumptions \(\lambda_1=\lambda_2\), \(\mu_1=\mu_2\), we determine the number of \(\gamma\)-bifurcations for each \(\beta\in(-1,\infty)\), and study the behavior of global \(\gamma\)-bifurcation branches in \([-1,0]\times H_r^1(\mathbb{R}^N) \times H_r^1(\mathbb{R}^N)\). Moreover, several results for \(\gamma=0\), such as priori bounds, are of independent interests, which are improvements of corresponding theorems in [T. Bartsch et al., Calc. Var. Partial Differ. Equ. 37, No. 3–4, 345–361 (2010; Zbl 1189.35074)] and [J. Wei and W. Yao, Commun. Pure Appl. Anal. 11, No. 3, 1003–1011 (2012; Zbl 1264.35237)].

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35B32 Bifurcations in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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