Existence of constrained minimizer for a quadratically coupled Schrödinger systems

ABSTRACT

In this paper, we consider the existence of solutions to a quadratically coupled Schrödinger systems $-\mathrm{\Delta }u={\mu }_{1}u+2\alpha uvin{R}^N,-\mathrm{\Delta }v={\mu }_{2}v+\alpha u^{2}in{R}^N$ under the condition ${\int }_{{\mathbb{R}}^N}{u}^2=a,{\int }_{{\mathbb{R}}^N}{v}^2=b.$ Here N=1, 2, $\alpha >0$ and a, b>0 are fixed. In the systems, ${\mu }_1$ and ${\mu }_2$ are unknown. We prove that there exists a solution $({\mu }_1,{\mu }_2,\tilde{u},\tilde{v})$ to the systems with ${\mu }_1<0$, ${\mu }_2<0$, and $\tilde{u},\tilde{v}\in C^2\left({\mathbb{R}}^N\right)$ being positive, radially symmetric, and decreasing with $r=|x|$. Our argument is heavily based on the rearrangement techniques.

KEYWORDS:

• Quadratically coupled Schrödinger systems
• solutions
• rearrangement techniques

AMS SUBJECT CLASSIFICATIONS:

• 35J50
• 35J55

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Partially supported by National Natural Science Foundation of China [grant numbers 11571209, 11671239, 11701346].