The Pohozaev Identity for the Fractional Laplacian
2014; Springer Science+Business Media; Volume: 213; Issue: 2 Linguagem: Inglês
10.1007/s00205-014-0740-2
ISSN1432-0673
AutoresXavier Ros‐Oton, Joaquim Serra,
Tópico(s)Nonlinear Differential Equations Analysis
ResumoIn 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.
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