Regularity and existence of positive solutions for a fractional system

Ran Zhuo; Yan Li

doi:10.3934/cpaa.2021168

Article Contents

2022, Volume 21, Issue 1: 83-100. Doi: 10.3934/cpaa.2021168

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Regularity and existence of positive solutions for a fractional system

Ran Zhuo^1, and
Yan Li^2, ,

1.
Department of Mathematics and Statistics, Huanghuai University, Zhumadian, Henan 463000, China
2.
Department of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China

^* Corresponding author
^* Corresponding author

Received: April 30, 2021

Revised: July 31, 2021

Early access: September 2021

Published: January 2022

The first author is partially supported by NSFC 11701207

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Abstract

We consider the nonlinear fractional elliptic system

$ \begin{equation*} \left\{\begin{array}{ll} (- \Delta)^{\frac{\alpha_1}{2}}u(x) = f(x, u, v), & \text{in}\, \, \, \Omega, \\ (- \Delta)^{\frac{\alpha_2}{2}}v(x) = g(x, u, v), & \text{in}\, \, \, \Omega, \\ u = v = 0, & \text{in}\, \, \, \mathbb{R}^n\setminus\Omega, \end{array} \right. \label{a-1.2} \end{equation*} $

where $ 0<\alpha_1, \alpha_2<2 $ and $ \Omega $ is a bounded domain with $ C^2 $ boundary in $ \mathbb{R}^n $. To overcome the technical difficulty due to the different fractional orders, we employ two distinct methods and derive the a priori estimates for $ 0<\alpha_1, \alpha_2<1 $ and $ 1<\alpha_1, \alpha_2 <2 $ respectively. Moreover, combining the a priori estimate with the topological degree theory, we prove the existence of positive solutions.

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Mathematics Subject Classification: Primary: 35B09, 35S15; Secondary: 35B44, 35B45, 35B65.

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