On critically coupled (s_1, s_2)-fractional system of Schr\"odinger equations with Hardy potential

TL;DR

This paper investigates the existence of bound and ground state solutions for a coupled fractional Schr"odinger system with Hardy potentials, employing variational methods and concentration-compactness principles.

Contribution

It establishes new existence results for ground states of a coupled fractional Schr"odinger system with Hardy potentials under specific conditions.

Findings

Existence of ground state solutions proven.

Application of mountain-pass theorem and concentration-compactness.

Conditions on parameters and functions ensuring solutions.

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.