# zbMATH — the first resource for mathematics

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.)
##### Keywords:
bifurcation; Schrödinger systems; positive solutions
Full Text: