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The Pohozaev identity for the fractional Laplacian. (English) Zbl 1361.35199

Summary: 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, \(s\in(0,1)\), \((-\Delta)^{s}\) is the fractional Laplacian in \(\mathbb{R}^n\), and \(\Omega\) 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 \(C^{\alpha}\) up to the boundary \(\partial\Omega\), where \(\delta(x) = \text{dist}(x,\partial\Omega)\). In the fractional Pohozaev identity, the function \(u/{\delta}^s|_{\partial{\Omega}}\) plays the role that \(\partial u/\partial\nu\) plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over \(\partial\Omega\)) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.

MSC:

35R11 Fractional partial differential equations
35J61 Semilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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