Partial regularity of the fractional Gelfand-Liouville system

Yeyao Hu; Yuan Li; Wen Yang

doi:10.3934/dcds.2023026

Article Contents

2023, Volume 43, Issue 7: 2721-2736. Doi: 10.3934/dcds.2023026

This issue Previous Article Global dynamics of evolution systems with asymptotic annihilation Next Article Observable Lyapunov irregular sets for planar piecewise expanding maps

Partial regularity of the fractional Gelfand-Liouville system

1.
School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, China
2.
School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
3.
School of Mathematics, Hunan University, Changsha 410082, China
4.
Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China

^*Corresponding author: Yeyao Hu
^*Corresponding author: Yeyao Hu

Received: May 2022

Revised: February 2023

Early access: March 2023

Published: July 2023

The first author is partially supported by NSFC grants No.12101612 and No.12171456. The second author is partially supported by Science and Technology Commission of Shanghai Municipality No.22DZ2229014 and China Postdoctoral Science Foundation No.2022M721164. The research of the third author is partially supported by National Key R & D Program of China 2022YFA1006800, NSFC grants No.12171456 and No.12271369.

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Abstract

We study the extremal solutions of fractional Dirichlet problem

$ \begin{equation} \nonumber \begin{cases} (-\Delta)^{s}u=\mu e^{v} &\mbox{in}\quad\Omega\\ (-\Delta)^{s}v=\lambda e^{u} &\mbox{in}\quad\Omega\\ u=v=0 &\mbox{on}\quad\mathbb{R}^{n}\setminus\Omega,\\ \end{cases} \end{equation} $

where $ s\in(0, 1) $, $ \Omega\subset\mathbb{R}^{n} $ is a bounded domain and $ \mu $ with $ \lambda $ are positive parameters. We combined Caffarelli-Silvestre extension technique and $ \varepsilon $-regularity theory to obtain that the Hausdorff dimension of the singular set does not exceed $ n-10s $. This result fills in the gap of partial regularity of the fractional Gelfand-Liouville system.

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Mathematics Subject Classification: Primary: 35B65; Secondary: 35R11.

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Table . Notations:

$X=(x,t)$		represent points in $\mathbb{R}^{n+1}_{+}=\mathbb{R}^{n}\times[0,\infty)$;
$B_{r}(x)$		the ball centered at x with radius $ r~ \mbox{in}~ \mathbb{R}^{n}, B_{r}:=B_{r}(0)$;
$B_{r}^{n+1}(x)$		the ball centered at x $\mbox{with radius}~ r~ \mbox{in}~ \mathbb{R}^{n+1}, B_{r}^{n+1}:=B_{r}^{n+1}(0)$;
$D_{r}(x)$		the intersection of $B_{r}^{n+1}(x)~ \mbox{and}~ \mathbb{R}_{+}^{n+1}, D_{r}:=D_{r}(0)$,
$f_k$		$f_k=\min\{f,k\}$,
$u_+$		$u_{+}=\max\{u,0\}$,
C		C will stand for positive universal constants which are allowed to vary among different formulas and even within the same lines.

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