## 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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### References:

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