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

Abstract

Consider the following system of double coupled Schrödinger equations arising from Bose-Einstein condensates etc.,

$\left\{ \begin{gathered} - \Delta u + u = \mu _1 u^3 + \beta uv^2 - \kappa v, \hfill \\ - \Delta v + v = \mu _2 v^3 + \beta u^2 v - \kappa u, \hfill \\ u \ne 0,v \ne 0,andu,v \in H^1 (\mathbb{R}^N ), \hfill \\ \end{gathered} \right. $

where µ₁, µ₂ 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 to the scalar equation −Δω + ω = ω ³, ω ∈ H ¹ _r (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 × H ¹ _r (ℝ^N) × H ¹ _r (ℝ^N) with parameter κ.