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

TL;DR

This paper investigates the existence, nonexistence, and bifurcation of solutions for a coupled Schrödinger system modeling Bose-Einstein condensates, using critical point theory and bifurcation analysis.

Contribution

It provides new existence and nonexistence results and constructs bifurcation branches for coupled Schrödinger equations with specific parameters.

Findings

Existence of positive solutions under certain parameter conditions.

Nonexistence results for some parameter regimes.

Identification of bifurcation points leading to solution branches.

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 $\kappa$.