Normalized solutions for a coupled Schrödinger system

Abstract

In the present paper, we prove the existence of solutions $(\lambda _1,\lambda _2,u,v)\in \mathbb {R}^2\times H^1(\mathbb {R}^3,\mathbb {R}^2)$ to systems of coupled Schrödinger equations

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 u^3+\beta uv^2\quad &{}\hbox {in}\;\mathbb {R}^3\\ -\Delta v+\lambda _2v=\mu _2 v^3+\beta u^2v\quad &{}\hbox {in}\;\mathbb {R}^3\\ u,v>0&{}\hbox {in}\;\mathbb {R}^3 \end{array}\right. } \end{aligned}$$

satisfying the normalization constraint $\int _{\mathbb {R}^3}u^2=a^2\quad \hbox {and}\;\int _{\mathbb {R}^3}v^2=b^2,$ which appear in binary mixtures of Bose–Einstein condensates or in nonlinear optics. The parameters $\mu _1,\mu _2,\beta >0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the case of fixed frequencies $\lambda _1,\lambda _2$. When the masses are prescribed, the standard approach to this problem is variational with $\lambda _1,\lambda _2$ appearing as Lagrange multipliers. Here we present a new approach based on the fixed point index in cones, bifurcation theory, and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $\beta$ in a large range. We also have a result about the nonexistence of positive solutions which shows that our existence theorem is almost optimal. Especially, if $\mu _1=\mu _2$ we prove that normalized solutions exist for all $\beta >0$ and all $a,b>0$.