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MiS Preprint
16/2008

Harnack's inequality for some nonlocal equations and application

Jerome Coville

Abstract

In this paper, we establish a Harnack's inequality for positive solutions of the nonlocal inhomogeneous problem $$\int_{\Omega}J\left(\frac{x-y}{g(y)}\right)\frac{u(y)}{g^n(y)}\, dy -a(x)u,$$ where $\Omega\subset\mathbb{R}^n$ is an open set, $J$ is a probability density with compact support and $g,b$ are positive bounded functions. For some particular $a(x)$, using the Harnack's Inequality, we also construct a positive solution of the above equation.

Received:
Jan 11, 2008
Published:
Jan 17, 2008
MSC Codes:
45A05, 47G10, 45M20, 47B34
Keywords:
Integral operator, Hanarck's inequality, Positive solution

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Jérôme Coville

Harnack's inequality for some nonlocal equations and application