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Existence and bifurcation of solutions for a double coupled system of Schrodinger equations

Consider the following system of double coupled Schrödinger equations arising from Bose-Einstein condensates etc., \begin{equation*} \left\{\begin{array}{l} -Δu + u =μ_1 u^3 + βuv^2- κv, -Δv + v =μ_2 v^3 + βu^2v- κu, u\neq0, v\neq0\ \hbox{and}\ u, v\in H^1(\R^N), \end{array} \right. \end{equation*}where $μ_1, μ_2$ are positive and fixed, $κ$ and $β$ 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 $-Δω+ω=ω^3$, $ω\in H_r^1(\R^N)$, we construct a synchronized solution branch to prove that for $β$ 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 $κ$.