Normalized Ground-State Solution for the Schrödinger–KdV System

Abstract

In this paper, we study the existence of ground-state solution for the coupled nonlinear Schrödinger–Korteweg–de Vries system

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda _{1}u=u^{3}+\beta uv&{}\quad \mathrm {in}\ \mathbb {R}^{N},\\ -\Delta v+\lambda _{2}v=\frac{1}{2}v^{2}+\frac{1}{2}\beta u^{2}&{} \quad \mathrm {in}\ \mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$

under the constrains

$$\begin{aligned} \int _{\mathbb {R}^{N}}|u|^2{\text {d}}x=a,\ \ \int _{\mathbb {R}^{N}}|v|^2{\text {d}}x=b,\\ \end{aligned}$$

where $N=1,2$, $\beta >0$ and $a, b>0$ are prescribed. In the system, the parameters $\lambda _{1}$ and $\lambda _{2}$ are unknown and will correspond to the Lagrange multipliers. Based on the variational method and rearrangement techniques, we prove that there exists a positive ground state solution for the system by the precompactness of the minimizing sequences, up to translation.