Regularity and Liouville theorem on an integral equation of Allen-Cahn type

Qinghua Chen; Yutian Lei

doi:10.3934/dcds.2024085

Article Contents

2025, Volume 45, Issue 1: 37-55. Doi: 10.3934/dcds.2024085

This issue Previous Article Optimal boundary regularity and a Hopf-type lemma for Dirichlet problems involving the logarithmic Laplacian Next Article Breakdown of homoclinic orbits to $ L_3 $: Nonvanishing of the Stokes constant

Regularity and Liouville theorem on an integral equation of Allen-Cahn type

Qinghua Chen^1, and
Yutian Lei^2, ,

1.
Department of Mathematics, Jinling Institute of Technology, China
2.
Ministry of Education Key Laboratory for NSLSCS, Nanjing Normal University, China

^*Corresponding author: Yutian Lei
^*Corresponding author: Yutian Lei

Received: February 2024

Revised: June 2024

Early access: June 2024

Published: January 2025

The first author was supported by PhD Research Start-up Fund of Jinling Institute of Technology [jit-b-202354], the second author was supported by NSFC [11871278].

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Abstract

In this paper, we studied an integral equation of the Allen-Cahn type

$ \begin{align*} u(x) = \overrightarrow{l} +C_{*}\int_{{\mathbb{R}}^n}\frac{u(y)(1-|u(y)|^2)|1-|u(y)|^2|^{p-2}} {|x-y|^{n-\alpha}}dy. \end{align*} $

Here, $ u:{\mathbb{R}}^n\rightarrow {\mathbb{R}}^k $, $ k \geq 1 $, $ n \geq 2 $, $ \alpha \in (0,n) $ and $ p-1>n/(n-\alpha) $. In addition, $ \overrightarrow{l}\in {\mathbb{R}}^k $ is a constant vector and $ C_{*} \neq 0 $ is a real constant. This equation is associated with the following fractional-order equation

$ (-\Delta)^{\frac{\alpha}{2}}u = u(1-|u|^2)|1-|u|^2|^{p-2}. $

Both of them play important roles in the study of phase transition in fractal media and the competition model of biological communities. We were concerned about two properties of those equations: One is the Lipschitz regularity of the bounded solutions; the other is the Liouville theorem. The regularity lifting lemma introduced by Ma-Chen-Li and the Pohozaev identity in integral form come into play here.

Keywords:

Mathematics Subject Classification: Primary: 45G05, 45E10; Secondary: 47N20, 35Q56, 35R11.

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