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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An integral equation on half space


Authors: Dongyan Li and Ran Zhuo
Journal: Proc. Amer. Math. Soc. 138 (2010), 2779-2791
MSC (2010): Primary 35J99, 45E10, 45G05
Published electronically: April 14, 2010
MathSciNet review: 2644892
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Abstract: Let $ R^n_+$ be the $ n$-dimensional upper half Euclidean space, and let $ \alpha$ be any real number satisfying $ 0<\alpha<n.$ In this paper, we consider the integral equation

$\displaystyle u(x)=\int_{R^n_+} (\dfrac{1}{\vert x-y\vert^{n-\alpha}}-\dfrac{1}{\vert x^*-y\vert^{n-\alpha}})u^\tau(y), u(x)>0, \forall x \in R_+^n,$ (1)

where $ \tau =\dfrac{n+\alpha}{n-\alpha}$, and $ x^*=(x_1,\cdots,x_{n-1},-x_n)$ is the reflection of the point $ x$ about the hyperplane $ x_n =0$. We use a new type of moving plane method in integral forms introduced by Chen, Li and Ou to establish the regularity and rotational symmetry of the solution of the above integral equation.


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Additional Information

Dongyan Li
Affiliation: College of Mathematics and Information Science, Henan Normal University, Henan, People’s Republic of China
Email: w408867388w@126.com

Ran Zhuo
Affiliation: College of Mathematics and Information Science, Henan Normal University, Henan, People’s Republic of China
Email: zhuoran1986@126.com

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10368-2
PII: S 0002-9939(10)10368-2
Keywords: Integral equations, regularity, method of moving planes, rotational symmetry, upper half space, monotonicity.
Received by editor(s): September 25, 2009
Published electronically: April 14, 2010
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.