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
modified on 12 Jun 2024


Accepted Paper

Inserted: 3 feb 2024
Last Updated: 12 jun 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:

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

$ (-\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}, $

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

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.