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Archive for Rational Mechanics and Analysis

, Volume 213, Issue 2, pp 587–628 | Cite as

The Pohozaev Identity for the Fractional Laplacian

  • Xavier Ros-OtonEmail author
  • Joaquim Serra
Article

Abstract

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem \({(-\Delta)^s u =f(u)}\) in \({\Omega, u\equiv0}\) in \({{\mathbb R}^n\backslash\Omega}\) . Here, \({s\in(0,1)}\) , (−Δ) s is the fractional Laplacian in \({\mathbb{R}^n}\) , and Ω is a bounded C 1,1 domain. To establish the identity we use, among other things, that if u is a bounded solution then \({u/\delta^s|_{\Omega}}\) is C α up to the boundary ∂Ω, where δ(x) = dist(x,∂Ω). In the fractional Pohozaev identity, the function \({u/\delta^s|_{\partial\Omega}}\) plays the role that ∂u/∂ν plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over ∂Ω) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.

Keywords

Weak Solution Viscosity Solution Bounded Solution Nonlocal Problem Nonexistence Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain

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