Normalized solutions for Schrödinger systems in dimension two

Abstract

In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger systems with exponential growth \begin{align*} \left\{ \begin{aligned} &-Δu+λ_{1}u=H_{u}(u,v), \quad \quad \hbox{in }\mathbb{R}^{2},\\ &-Δv+λ_{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 Carathéodory function $H$ with $H_u,H_v$ have exponential growth in ${\mathbb{R}}^2$. Our main results are totally new for Schrödinger systems in ${\mathbb{R}}^2$. Using the Pohozaev manifold and variational methods, we establish the existence of normalized solutions to the above problem.

arXiv admin note: text overlap with arXiv:2102.03001 by other authors