Normalized solutions for Schr\"{o}dinger systems in dimension two

Abstract

In this paper, we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger systems with exponential growth \begin{align*} \left\{ \begin{aligned} &-\Delta u+\lambda_{1}u=H_{u}(u,v), \quad \quad \hbox{in }\mathbb{R}^{2},\\ &-\Delta v+\lambda_{2} v=H_{v}(u,v), \quad \quad \hbox{in }\mathbb{R}^{2},\\ &\int_{\mathbb{R}^{2}}|u|^{2}dx=a^{2},\quad \int_{\mathbb{R}^{2}}|v|^{2}dx=b^{2}, \end{aligned} \right. \end{align*} where $a,b>0$ are prescribed, $\lambda_{1},\lambda_{2}\in \mathbb{R}$ and the functions $H_{u},H_{v}$ are partial derivatives of a Carath\'{e}odory function $H$ with $H_{u},H_{v}$ have exponential growth in $\mathbb{R}^{2}$. Our main results are totally new for Schr\"{o}dinger systems in $\mathbb{R}^{2}$. Using the Pohozaev manifold and variational methods, we establish the existence of normalized solutions to the above problem.