Calculus of Variations and Geometric Measure Theory

R. Kumar - T. Mukherjee - A. Sarkar

On critically coupled (s_1, s_2)-fractional system of Schrödinger equations with Hardy potential

created by kumar2 on 03 Feb 2024


Accepted Paper

Inserted: 3 feb 2024
Last Updated: 3 feb 2024

Journal: Differential and Integral Equations 2023
Year: 2023

ArXiv: 2210.08260 PDF


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}{
{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~~}{
{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 $\Lambda_{N,s_{i}} = 2 \pi^{N/2} \frac{\Gamma^{2}(\frac{N+2s_i}{4}) \Gamma(\frac{N+2s_i}{2})}{\Gamma^{2}(\frac{N-2s_i}{4}) ~
}, (i=1,2)$. 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.