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

 

Regularity of nonlinear equations for fractional Laplacian


Authors: Aliang Xia and Jianfu Yang
Journal: Proc. Amer. Math. Soc. 141 (2013), 2665-2672
MSC (2010): Primary 35J25, 47G30, 35B45, 35J70
Published electronically: April 26, 2013
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove that any $ H^s(\Omega )$ solution $ u$ of the problem

$\displaystyle (-\Delta )^s u=f(u)\,\, {\rm in}\,\,\Omega , \quad u = 0 \,\, {\rm on}\,\, \partial \Omega ,$ (1)

belongs to $ L^\infty (\Omega )$ for the nonlinearity of $ f(t)$ being subcritical and critical. This implies that the solution $ u$ is classical if $ f(t)$ is $ C^{1,\gamma }$ for some $ 0<\gamma <1$.

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Additional Information

Aliang Xia
Affiliation: Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, People’s Republic of China
Email: xiaaliang@sina.com

Jianfu Yang
Affiliation: Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, People’s Republic of China
Email: jfyang\textunderscore2000@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11734-X
PII: S 0002-9939(2013)11734-X
Keywords: Regularity, $L^\infty$ bounds, fractional Laplacian
Received by editor(s): February 6, 2011
Published electronically: April 26, 2013
Additional Notes: The second author is supported by National Natural Sciences Foundations of China, grant No. 11271170, and the GAN PO 555 program of Jiangxi.
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.