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 $$\{\begin{array}{ll}-\Delta u+{\lambda }_{1}u={\mu }_1{u}^3+\beta u{v}^2& in{\mathbb{R}}^3\\ -\Delta v+{\lambda }_{2}v={\mu }_2{v}^3+\beta u^{2}v& in{\mathbb{R}}^3\\ u,v>0& in{\mathbb{R}}^3\end{array}$$ satisfying the normalization constraint ${\int }_{{\mathbb{R}}^3}{u}^2=a^{2}and{\int }_{{\mathbb{R}}^3}{v}^2=b^2,$ 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 fixed frequency case. And 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 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 give a result about the nonexistence of positive solutions. From which one can see that our existence theorem is almost the best. Especially, if ${\mu }_1={\mu }_2$ we prove that normalized solutions exist for all $\beta >0$ and all $a,b>0$.

27 pages, 1 figure

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