preprint

Inserted: 3 feb 2024
Last Updated: 12 jun 2024

Year: 2023

Abstract:

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:

$ (-\Delta)^{s_{1}} u - \lambda_{1} \frac{u}{
x
^{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}{
x
^{2s_{2}}} - v^{2_{s_{2}}^{*}-1} = \nu h(x) u^{2} \quad \mbox{in} ~ \mathbb{R}^{N},$

$u,v >0 \quad \mbox{in} ~ \mathbb{R}^{N} \setminus \{0\},$

where $s_{1},s_{2} \in (0,1)~\text{and}~\lambda_{i}\in (0, \Lambda_{N,s_{i}})$ with $\Lambda_{N,s_{i}} = 2 \pi^{N/2} \frac{\Gamma^{2}(\frac{N+2s_i}{4}) \Gamma(\frac{N+2s_i}{2})}{\Gamma^{2}(\frac{N-2s_i}{4}) ~
\Gamma(-s_{i})
}, (i=1,2)$. 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.