Normalized solutions to Schr\"{o}dinger systems with linear and nonlinear couplings

TL;DR

This paper investigates normalized solutions to coupled Schrödinger systems with linear and nonlinear interactions, proving existence of radially symmetric solutions in subcritical and supercritical cases using variational methods.

Contribution

It introduces new existence results for normalized solutions of Schrödinger systems with both linear and nonlinear couplings, employing variational principles in different critical regimes.

Findings

Existence of normalized radially symmetric solutions in the subcritical case.

Existence of positive solutions in the supercritical case for specific exponents.

Application of variational methods to coupled Schrödinger systems with fixed L^2 norms.

Abstract

In this paper, we study important Schr\"{o}dinger systems with linear and nonlinear couplings \begin{equation}\label{eq:diricichlet} \begin{cases} -\Delta u_1-\lambda_1 u_1=\mu_1 |u_1|^{p_1-2}u_1+r_1\beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+\kappa (x)u_2~\hbox{in}~\mathbb{R}^N,\\ -\Delta u_2-\lambda_2 u_2=\mu_2 |u_2|^{p_2-2}u_2+r_2\beta |u_1|^{r_1}|u_2|^{r_2-2}u_2+\kappa (x)u_1~ \hbox{in}~\mathbb{R}^N,\\ u_1\in H^1(\mathbb{R}^N), u_2\in H^1(\mathbb{R}^N),\nonumber \end{cases} \end{equation} with the condition $$\int_{\mathbb{R}^N} u_1^2=a_1^2, \int_{\mathbb{R}^N} u_2^2=a_2^2,$$ where $N\geq 2$, $\mu_1,\mu_2,a_1,a_2>0$, $\beta\in\mathbb{R}$, $2<p_1,p_2<2^*$, $2<r_1+r_2<2^*$, $\kappa(x)\in L^{\infty}(\mathbb{R}^N)$ with fixed sign and $\lambda_1,\lambda_2$ are Lagrangian multipliers. We use Ekland variational principle to prove this system has a normalized radially symmetric solution for $L^2-$subcritical case when $N\geq 2$, and use minimax method to prove this system has a normalized radially symmetric positive solution for $L^2-$supercritical case when $N=3$, $p_1=p_2=4,\ r_1=r_2=2$.