Existence and asymptotic behavior of normalized ground states for Sobolev critical Schr\"odinger systems

TL;DR

This paper investigates the existence and asymptotic behavior of normalized ground states in Sobolev critical Schrödinger systems, establishing conditions for existence based on parameters and analyzing the limits as coupling constants vary.

Contribution

It provides new existence criteria for normalized ground states in critical Schrödinger systems and describes their asymptotic behavior as the coupling parameter approaches zero or infinity.

Findings

Normalized ground states do not exist for negative coupling.

Existence of ground states depends on the sign and magnitude of the coupling constant.

Asymptotic analysis reveals the behavior of solutions as the coupling parameter tends to zero or infinity.

Abstract

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schr\"odinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-\delta u+\lambda_1 u=|u|^{2^*-2}u+{\nu\alpha} |u|^{\alpha-2}|v|^\beta u,\quad \text{in }\mathbb{R}^N, &-\delta v+\lambda_2 v=|v|^{2^*-2}v+{\nu\beta} |u|^\alpha |v|^{\beta-2}v,\quad \text{in }\mathbb{R}^N, &\int u^2=a^2,\;\;\; \int v^2=b^2, \end{aligned} \right. \end{equation*} where $N=3,4$, $\alpha,\beta>1$, $2<\alpha+\beta<2^*=\frac{2N}{N-2}$. We prove that a normalized ground state does not exist for $\nu<0$. When $\nu>0$ and $\alpha+\beta\le 2+\frac{4}{N}$, we show that the system has a normalized ground state solution for $0<\nu<\nu_0$, the constant $\nu_0$ will be explicitly given. In the case $\alpha+\beta>2+\frac{4}{N}$ we prove the existence of a threshold $\nu_1\ge 0$ such that a normalized ground state solution exists for $\nu>\nu_1$, and does not exist for $\nu<\nu_1$. We also give conditions for $\nu_1=0$. Finally we obtain the asymptotic behavior of the minimizers as $\nu\to0^+$ or $\nu\to+\infty$.