Normalized solutions of two-component nonlinear Schrödinger equations with linear couples

ABSTRACT

In this paper, we focus on the following nonlinear Schrödinger equations with linear couples $\{\begin{array}{ll}{\displaystyle -\mathrm{\Delta }u+V_1(x)u+{\lambda }_{1}u={\mu }_1{\int }_{{\mathbb{R}}^3}\frac{|u(y){|}^p}{|x-y|}\mathrm{d}y|u{|}^{p-2}u+\mathrm{\beta v}}& \mathrm{in}{\mathbb{R}}^3,\\ {\displaystyle -\mathrm{\Delta v}+V_2(x)v+{\lambda }_{2}v={\mu }_2{\int }_{{\mathbb{R}}^3}\frac{|v(y){|}^q}{|x-y|}\mathrm{d}y|v{|}^{q-2}v+\mathrm{\beta u}}& \mathrm{in}{\mathbb{R}}^3,\\ {\displaystyle {\int }_{{\mathbb{R}}^3}|u{|}^2\mathrm{d}x=a,{\int }_{{\mathbb{R}}^3}|v{|}^2\mathrm{d}x=b,}& \end{array}$ where $\frac{5}{3}<p,q<\frac{7}{3}$, ${\mu }_1,{\mu }_2>0$, $a,b\ge 0$, $\beta \in \mathbb{R}\setminus \{0\}$, ${\lambda }_1,{\lambda }_2\in \mathbb{R}$ are Lagrange multipliers and $V_1(x),V_2(x):{\mathbb{R}}^3\to \mathbb{R}$ are trapping potentials. We prove the existence of the solutions with prescribed $L^2(\mathbb{R})$-norm with trivial trapping potentials and nontrivial trapping potentials by applying the rearrangement inequalities.

KEYWORDS:

• Nonlinear Schrödinger systems
• normalized solution
• global minimizer
• potential

AMS SUBJECT CLASSIFICATION:

• 35-02

Additional information

Funding

Jiamo Li and Zifei Shen were supported by National Natural Science Foundation of China [No. 12071438].