An integral equation involving Bessel potentials on half space

Yonggang Zhao; Mingxin Wang

doi:10.3934/cpaa.2015.14.527

Article Contents

2015, Volume 14, Issue 2: 527-548. Doi: 10.3934/cpaa.2015.14.527

This issue Previous Article A note on the Monge-Kantorovich problem in the plane Next Article Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force

An integral equation involving Bessel potentials on half space

Yonggang Zhao^1, and
Mingxin Wang^2,

1.
Department of Mathematics, Harbin Institute of Technology, Harbin 150080
2.
Science Research Center, Harbin Institute of Technology, Harbin, 150080

Received: January 31, 2014

Revised: August 31, 2014

Published: February 2015

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Abstract

In this article we consider the following integral equation involving Bessel potentials on a half space $\mathbb{R}^n_+ $: \begin{eqnarray} u(x)=\int_{ \mathbb{R}^n_+ }\{g_\alpha(x-y)-g_\alpha(\bar x-y)\} u^\beta(y) dy,\;\;x\in \mathbb{R}^n_+, \end{eqnarray} where $\alpha>0$, $\beta>1$, $\bar x$ is the reflection of $x$ about $x_n=0$, and $g_\alpha(x)$ denotes the Bessel kernel. We first enhance the regularity of positive solutions for the integral equation by regularity-lifting-method, which has been extensively used by many authors. Then, employing the method of moving planes in integral forms, we demonstrate that there is no positive solution for the integral equation.

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Mathematics Subject Classification: Primary: 45E10; Secondary: 35G30, 35J60.

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