Calculus of Variations and Geometric Measure Theory

R. Kumar - T. Mukherjee - A. Sarkar

On a fractional system of NLS-KDV equations with Hardy potentials

created by kumar2 on 03 Feb 2024



Inserted: 3 feb 2024
Last Updated: 3 feb 2024

Year: 2023

ArXiv: 2309.09536 PDF


In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of nonlinear Schr\"odinger-Korteweg-De Vries (NLS-KdV, in short) equations with Hardy potentials: \begin{equation} \left\{ \begin{aligned} (-\Delta){s{1}} u - \lambda{1} \frac{u}{
{2s{1}}} - u{2{s{1}}{
}-1} &= 2\nu h(x) u{}v{} & \quad \mbox{in} ~ \mathbb{R}{N}, (-\Delta){s{2}} v - \lambda{2} \frac{v}{
{2s{2}}} - v{2{s{2}}{}-1} &= \nu h(x) u{2} & \quad \mbox{in} ~ \mathbb{R}{N}, u,v >0 \quad \mbox{in} ~ \mathbb{R}{N} \setminus \{0\}, \end{aligned} \right. \end{equation} where $s_{1},s_{2} \in (0,1)~\text{and}~\lambda_{i}\in (0, \Lambda_{N,s_{i}})$ with $\Lambda_{N,s_{i}} = 2 \pi^{N/2} \frac{\Gamma^{2}(\frac{N+2s_i}{4}) \Gamma(\frac{N+2s_i}{2})}{\Gamma^{2}(\frac{N-2s_i}{4}) ~
}, (i=1,2)$. By imposing certain assumptions on the parameter $\nu$ and on the function $h$, we obtain ground-state solutions using the concentration-compactness principle and the mountain-pass theorem.