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

Abstract

In this paper, we study the coupled Schrödinger-KdV system \begin{align*} \begin{cases} -Δu +λ_1 u=u^3+βuv~~&\text{in}~~\mathbb{R}^{3}, \\-Δv +λ_2 v=\frac{1}{2}v^2+\frac{1}{2}βu^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, ${\hat{I}}^2>0$, and the frequencies $\hat{I}{»}_1,\hat{I}{»}_2$ 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.

17 pages, accepted by Z. Angew. Math. Phys