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.


35R11 Fractional partial differential equations
35J61 Semilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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[1] Andrews G.E., Askey R., Roy R.: Special Functions. Cambridge University Press, Cambridge (2000) · Zbl 1075.33500
[2] Berndt B.C.: Ramanujan’s Notebooks, Part II. Springer, Berlin (1989) · Zbl 0716.11001
[3] Bogdan, K.; Grzywny, T.; Ryznar, M., Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab., 38, 1901-1923, (2010) · Zbl 1204.60074
[4] Brandle, C.; Colorado, E.; Pablo, A., A concave-convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinb. Sect. A, 143, 39-71, (2013) · Zbl 1290.35304
[5] Cabré, X.; Cinti, E., Sharp energy estimates for nonlinear fractional diffusion equations, Calc. Var. Partial Differ. Equ., 49, 233-269, (2014) · Zbl 1282.35399
[6] Cabré, X.; Tan, J., Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224, 2052-2093, (2010) · Zbl 1198.35286
[7] Caffarelli, L.; Roquejoffre, J.M.; Sire, Y., Variational problems in free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12, 1151-1179, (2010) · Zbl 1221.35453
[8] Caffarelli, L.; Silvestre, L., Regularity theory for fully nonlinear integro-differential equations, Commun. Pure Appl. Math., 62, 597-638, (2009) · Zbl 1170.45006
[9] Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32, 1245-1260, (2007) · Zbl 1143.26002
[10] Chen, W.; Li, C.; Ou, B., Classification of solutions to an integral equation, Commun. Pure Appl. Math., 59, 330-343, (2006) · Zbl 1093.45001
[11] Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992) · Zbl 0804.28001
[12] Fall, M.M.; Weth, T., Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263, 2205-2227, (2012) · Zbl 1260.35050
[13] Getoor, R.K., First passage times for symmetric stable processes in space, Trans. Am. Math. Soc., 101, 75-90, (1961) · Zbl 0104.11203
[14] de Pablo, A., Sánchez, U.: Some Liouville-type results for a fractional equation. Preprint · Zbl 1093.45001
[15] Pohozaev, S.I., On the eigenfunctions of the equation δ\(u\) + λ f(\(u\)) = 0, Dokl. Akad. Nauk SSSR, 165, 1408-1411, (1965) · Zbl 0141.30202
[16] Ros-Oton, X.; Serra, J., Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350, 505-508, (2012) · Zbl 1273.35301
[17] Ros-Oton, X.; Serra, J., The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101, 275-302, (2014) · Zbl 1285.35020
[18] Servadei, R.; Valdinoci, E., Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389, 887-898, (2012) · Zbl 1234.35291
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