The Pohozaev Identity for the Fractional Laplacian

By Xavier Ros-Oton, J. Serra. 2012

369 citations recorded by the source corpus.

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $${(-\Delta)^s u =f(u)}$$(-Δ)su=f(u) in $${\Omega, u\equiv0}$$Ω,u≡0 in $${{\mathbb R}^n\backslash\Omega}$$Rn\Ω . Here, $${s\in(0,1)}$$s∈(0,1) , (−Δ)s is the fractional Laplacian in $${\mathbb{R}^n}$$Rn , and Ω is a bounded C1,1 domain. To establish the identity we use, among other things, that if u is a bounded solution then $${u/\delta^s|_{\Omega}}$$u/δs|Ω is Cα up to the boundary ∂Ω, where δ(x) = dist(x,∂Ω). In the fractional Pohozaev identity, the function $${u/\delta^s|_{\partial\Omega}}$$u/δs|∂Ω 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/292265969fd260bd883543666826e1a7668903aa

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Citations in the corpus

Local Elliptic Regularity for the Dirichlet Fractional Laplacian (2017, methodology): “One can see that the latter property is not true by applying the Pohozaev identity obtained in [26] to the eigenfunctions of the Dirichlet fractional Laplacian.”
The Nirenberg problem and its generalizations: a unified approach (2014, background): “mple blow up points, we need an auxiliary bound in the below, and a Pohozaev type identity in Proposition 2.15. Pohozaev type identities for fractional Laplacian and some applications can be found in [95, 96]. Lemma 2.14. Under the hypotheses of Proposition 2.11 with r¯= 2, and in addition that xi →”
On fractional Choquard equations (2014, background): “The first complete study of Pohožaev identities and nonexistence results in star-shaped bounded domains for equations involving the fractional Laplacian and a local nonlinearity was done in [24, 25].”
A critical fractional equation with concave-convex power nonlinearities (2013, methodology): “This has been proved in [24,36] using a Pohozaev identity for the operator ( (cid”
A Global Compactness type result for Palais-Smale sequences in fractional Sobolev spaces (2014, background): “[15] X. Ros-Oton, J. Serra: The Pohozaev identity for the fractional Laplacian.”
On critical systems involving fractional Laplacian (2017, background): “Before going on, we need to establish the following Pohozaev identity (see [8] for the scalar equation case).”
Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian (2012, neutral): “1 in [22]).”
Fractional Orlicz-Sobolev embeddings (2020, background): “A touch of recent contributions in this connection is furnished by [7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 41, 44, 45, 51, 52, 53, 58, 59, 61, 62, 63, 64, 65, 68, 70, 71, 77, 76, 78, 83, 86].”
The Calderón problem for variable coefficients nonlocal elliptic operators (2017, background): “For A(x) being an identity matrix, the operator L becomes the well-known fractional Laplacian operator (−∆), and the detailed study about the (−∆) is available in [3, 6, 7, 8, 38, 39, 41, 42, 43].”
Existence results for fractional p-Laplacian problems via Morse theory (2014, neutral): “In the linear case p = 2, (RC) holds with ã = s due to the results of [39, 40].”
Nonlocal problems with critical Hardy nonlinearity (2017, background): “The latter is available only in the case p = 2 due to [23], leading in a standard way to the above mentioned non-existence result.”
Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials (2016, background): “As a matter of fact, the literature on fractional operators and their applications to partially differential equations is quite large, here we would like to mention a few, see for instance [2, 11, 12, 26, 27, 35].”