Normalized solutions for a coupled Schrödinger system

Abstract

In the present paper, we prove the existence of solutions $(λ_1,λ_2,u,v)\in\mathbb{R}^2\times H^1(\mathbb{R}^3,\mathbb{R}^2)$ to systems of coupled Schrödinger equations $$ \begin{cases} -Δu+λ_1u=μ_1 u^3+βuv^2\quad &\hbox{in}\;\mathbb{R}^3\\ -Δv+λ_2v=μ_2 v^3+βu^2v\quad&\hbox{in}\;\mathbb{R}^3\\ u,v>0&\hbox{in}\;\mathbb{R}^3 \end{cases} $$ satisfying the normalization constraint $ \displaystyle\int_{\mathbb{R}^3}u^2=a^2\quad\hbox{and}\;\int_{\mathbb{R}^3}v^2=b^2, $ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics. The parameters $μ_1,μ_2,β>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the fixed frequency case. And when the masses are prescribed, the standard approach to this problem is variational with $λ_1,λ_2$ appearing as Lagrange multipliers. Here we present a new approach based on bifurcation theory and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $β$ in a large range. We also give a result about the nonexistence of positive solutions. From which one can see that our existence theorem is almost the best. Especially, if $μ_1=μ_2$ we prove that normalized solutions exist for all $β>0$ and all $a,b>0$.