Accepted Paper
Inserted: 3 feb 2024
Last Updated: 3 feb 2024
Journal: Differential and Integral Equations 2023
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 Schr\"{o}dinger
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} = \nu \alpha h(x) u{\alpha-1}v{\beta} & \quad \mbox{in} ~
\mathbb{R}{N}, (-\Delta){s{2}} v - \lambda{2} \frac{v~~}{
x
{2s{2}}} -
v{2{s{2}}{}-1} = \nu \beta h(x) u{\alpha}v{\beta-1} & \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 parameters and on the function
h, we obtain ground-state solutions using the concentration-compactness
principle and the mountain-pass theorem.