Normalized solutions for a coupled Schr\"odinger system

Thomas Bartsch Mathematisches Institut, Justus-Liebig-Universit\"at Giessen, Arndtstrasse 2, 35392 Giessen, Germany Xuexiu Zhong South China Research Center for Applied Mathematics and Interdisciplinary Studies,South China Normal University, Guangzhou 510631, China. Wenming Zou Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China.

Analysis of PDEs mathscidoc:2006.03002

Math. Ann. , 2020
In the present paper, we prove the existence of solutions $(\lambda_1,\lambda_2,u,v)\in\R^2\times H^1(\R^3,\R^2)$ to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1u=\mu_1 u^3+\beta uv^2\quad &\hbox{in}\;\R^3\\ -\Delta v+\lambda_2v=\mu_2 v^3+\beta u^2v\quad&\hbox{in}\;\R^3\\ u,v>0&\hbox{in}\;\R^3 \end{cases} $$ satisfying the normalization constraint $ \displaystyle\int_{\R^3}u^2=a^2\quad\hbox{and}\;\int_{\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 $a,b>0$.
Schr\"odinger system; self-focusing; attractive interaction; solitary wave; normalized solution; global bifurcation; fixed point index in cones
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  title={Normalized solutions for a coupled Schr\"odinger system},
  author={Thomas Bartsch, Xuexiu Zhong, and Wenming Zou},
  booktitle={Math. Ann. },
Thomas Bartsch, Xuexiu Zhong, and Wenming Zou. Normalized solutions for a coupled Schr\"odinger system. 2020. In Math. Ann. .
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