An integral equation on half space
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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 \begin{equation} u(x)=\int _{R^n_+} (\dfrac {1}{|x-y|^{n-\alpha }}-\dfrac {1}{|x^*-y|^{n-\alpha }})u^\tau (y), u(x)>0, \forall x \in R_+^n, \end{equation} 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.References
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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
- Received by editor(s): September 25, 2009
- Published electronically: April 14, 2010
- Communicated by: Matthew J. Gursky
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 2779-2791
- MSC (2010): Primary 35J99, 45E10, 45G05
- DOI: https://doi.org/10.1090/S0002-9939-10-10368-2
- MathSciNet review: 2644892