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The Pohozaev identity for the fractional Laplacian

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10.1007/s00205-014-0740-2
 
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hdl:2117/23348

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Ros Oton, XavierMés informació
Serra Montolí, Joaquim
Document typeArticle
Defense date2014-08-01
Rights accessOpen Access
All rights reserved. This work is protected by the corresponding intellectual and industrial property rights. Without prejudice to any existing legal exemptions, reproduction, distribution, public communication or transformation of this work are prohibited without permission of the copyright holder
Abstract
In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem (-Delta)(s) u = f(u) in Omega, u equivalent to 0 in R-n\Omega. Here, s is an element of (0, 1), (-Delta)(s) is the fractional Laplacian in R-n, and Omega is a bounded C-1,C-1 domain. To establish the identity we use, among other things, that if u is a bounded solution then u/delta(s)vertical bar(Omega) is C-alpha up to the boundary partial derivative Omega, where delta(x) = dist(x, partial derivative Omega). In the fractional Pohozaev identity, the function u/delta(s)vertical bar(partial derivative Omega) plays the role that partial derivative u/partial derivative nu plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over partial derivative Omega) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.
CitationRos, X.; Serra, J. The Pohozaev identity for the fractional Laplacian. "Archive for rational mechanics and analysis", 01 Agost 2014, vol. 213, núm. 2, p. 587-628. 
URIhttp://hdl.handle.net/2117/23348
DOI10.1007/s00205-014-0740-2
ISSN0003-9527
Publisher versionhttp://link.springer.com/article/10.1007%2Fs00205-014-0740-2
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