分数阶非线性薛定谔系统规范解的存在性

Existence of Normalized Solution for Fractional Nonlinear Schrödinger Systems

摘要: 本文主要研究了一类分数阶非线性薛定谔系统 在$H^s\left(R^N\right)\times H^s\left(R^N\right)$ 中的规范解的存在性，且解满足${\displaystyle {\int }_{R^N}{\left|u_1\right|}^2\text{d}x}=a_1,{\displaystyle {\int }_{R^N}{\left|u_2\right|}^2\text{d}x}=a_2.$ 其中规定$s\in \left(0,1\right),a_1,a_2>0,{\lambda }_1,{\lambda }_2\in R$ 是以拉格朗日乘子出现的未知参数，$h_i:R^N\to \left[0,\infty \right)$ 为有界连续函数。考虑$f_i={\left|u_i\right|}^{p_i-2}{u}_i,i=1,2$ 且$F\left(u_1,u_2\right)=\omega {\left|u_1\right|}^{r_1}{\left|u_2\right|}^{r_2}$ ，$\omega ,r_1,r_2$ 是正的常数。

Abstract: This article mainly studies a class of fractional nonlinear Schrödinger coupled systems with the existence of the normalized solution in $H^s\left(R^N\right)\times H^s\left(R^N\right)$ , and the solution satisfies${\displaystyle {\int }_{R^N}{\left|u_1\right|}^2\text{d}x}=a_1,{\displaystyle {\int }_{R^N}{\left|u_2\right|}^2\text{d}x}=a_2.$ Among them, it is specified that $s\in \left(0,1\right),a_1,a_2>0,{\lambda }_1,{\lambda }_2\in R$ is a Lagrange multiplier; $h_i:R^N\to \left[0,\infty \right)$ is a bounded continuous function. Consider $f_i={\left|u_i\right|}^{p_i-2}{u}_i,i=1,2$ and $F\left(u_1,u_2\right)=\omega {\left|u_1\right|}^{r_1}{\left|u_2\right|}^{r_2}$ , where $\omega ,r_1,r_2$ are positive constant.