Fundamental solutions of a class of homogeneous integro-differential elliptic equations

Yi Cao; Jianhua Wu; Lihe Wang

doi:10.3934/dcds.2019053

Article Contents

2019, Volume 39, Issue 3: 1237-1256. Doi: 10.3934/dcds.2019053

This issue Previous Article Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions Next Article Fractional equations with indefinite nonlinearities

Fundamental solutions of a class of homogeneous integro-differential elliptic equations

1.
College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119, China
2.
Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China
3.
Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA

* Corresponding author: NSFC grant 11271236
* Corresponding author: NSFC grant 11271236

Received: March 31, 2017

Revised: January 31, 2018

Published: December 2018

The first author is supported by NSFC grant 11126201, 11671243

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Abstract

In this paper, we study a class of integro-differential elliptic operators $L_{σ}$ with kernel $k(y) = a(y)/|y|^{d+σ}$, where $d≥2, σ∈(0,2)$, and the positive function $a(y)$ is homogenous and bounded. By using a purely analytic method, we construct the fundamental solution $Φ$ of $L_{σ}$ if $a(y)$ satisfies a natural cancellation assumption and $|a(y)-1|$ is small. Furthermore, we show that the fundamental solution $Φ$ is $-α^{*}$ homogeneous and Lipschitz continuous, where the constant $α^{*}∈(0,d)$. A Liouville-type theorem demonstrates that the fundamental solution $Φ$ is the unique nontrivial solution of $L_{σ}u = 0$ in $\mathbb{R}^{d}\setminus\{0\}$ that is bounded from below.

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Mathematics Subject Classification: Primary: 45K05; Secondary: 35R09.

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