Normalized ground state solutions of Schr\"odinger-KdV system in $\mathbb{R}^3$

TL;DR

This paper establishes the existence of normalized ground state solutions for a coupled Schr"odinger-KdV system in three-dimensional space, using a novel constrained minimization approach despite the system's supercritical growth.

Contribution

The paper introduces a new constraint minimization method to prove the existence of normalized ground states for the Schr"odinger-KdV system with $L^2$-supercritical growth.

Findings

Existence of a local minimum solution to the system.

Existence of normalized ground state solutions.

Application of a novel minimization approach for supercritical systems.

Abstract

In this paper, we study the coupled Schr\"odinger-KdV system \begin{align*} \begin{cases} -\Delta u +\lambda_1 u=u^3+\beta uv~~&\text{in}~~\mathbb{R}^{3}, \\-\Delta v +\lambda_2 v=\frac{1}{2}v^2+\frac{1}{2}\beta u^2~~&\text{in}~~\mathbb{R}^{3} \end{cases} \end{align*} subject to the mass constraints \begin{equation*} \int_{\mathbb{R}^{3}}|u|^2 dx=a,\quad \int_{\mathbb{R}^{3}}|v|^2 dx=b, \end{equation*} where $a, b>0$ are given constants, $\beta>0$, and the frequencies $\lambda_1,\lambda_2$ arise as Lagrange multipliers. The system exhibits $L^2$-supercritical growth. Using a novel constraint minimization approach, we demonstrate the existence of a local minimum solution to the system. Furthermore, we establish the existence of normalized ground state solutions.