preprint
Inserted: 3 feb 2024
Last Updated: 3 feb 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:
\begin{equation}
\left\{
\begin{aligned}
(-\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\},
\end{aligned}
\right. \end{equation}
where $s_{1},s_{2} \in (0,1)~\text{and}~\lambda_{i}\in (0,
\Lambda_{N,s_{i}})$ with $\Lambda_{N,s_{i}} = 2 \pi^{N/2}
\frac{\Gamma^{2}(\frac{N+2s_i}{4})
\Gamma(\frac{N+2s_i}{2})}{\Gamma^{2}(\frac{N-2s_i}{4}) ~
\Gamma(-s_{i})
},
(i=1,2)$. 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.