Fractional Laplacian: Pohozaev identity and nonexistence results

Xavier Ros-Oton; Joaquim Serra

doi:10.1016/j.crma.2012.05.011

Partial Differential Equations

Fractional Laplacian: Pohozaev identity and nonexistence results
[Laplacien fractionnaire : identité de Pohozaev et résultats de nonexistence]

Présenté par : Haïm Brézis

Xavier Ros-Oton ¹ ; Joaquim Serra ¹

¹ Universitat Politècnica de Catalunya, Departament Matemàtica Aplicada I, Avda. Diagonal 647, 08028 Barcelona, Spain

Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 505-508.

Résumé
Abstract

Dans cette Note, nous présentons lʼidentité de Pohozaev pour le Laplacien fractionnaire. Comme conséquence de cette identité, nous prouvons la non-existence de solutions non triviales pour les problèmes semi-linéaires avec nonlinéarité sur-critique dans des domaines étoilés.

In this Note we present the Pohozaev identity for the fractional Laplacian. As a consequence of this identity, we prove the nonexistence of nontrivial bounded solutions to semilinear problems with supercritical nonlinearities in star-shaped domains.

Reçu le : 2012-05-03
Accepté le : 2012-05-25
Publié le : 2012-05-01

DOI : 10.1016/j.crma.2012.05.011

Affiliations des auteurs :

Xavier Ros-Oton ¹ ; Joaquim Serra ¹

¹ Universitat Politècnica de Catalunya, Departament Matemàtica Aplicada I, Avda. Diagonal 647, 08028 Barcelona, Spain

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Xavier Ros-Oton; Joaquim Serra. Fractional Laplacian: Pohozaev identity and nonexistence results. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 505-508. doi : 10.1016/j.crma.2012.05.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.011/

Bibliographie
Cité par

[1] M.M. Fall; T. Weth Nonexistence results for a class of fractional elliptic boundary value problems | arXiv

[2] S.I. Pohozaev On the eigenfunctions of the equation $Δ u + λ f (u) = 0$ , Dokl. Akad. Nauk SSSR, Volume 165 (1965), pp. 1408-1411

[3] X. Ros-Oton, J. Serra, The Pohozaev identity for the fractional Laplacian, forthcoming.

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Guillaume Penent; Nicolas Privault Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls, Communications on Pure and Applied Analysis, Volume 22 (2023) no. 8, pp. 2646-2660 | DOI:10.3934/cpaa.2023081 | Zbl:1522.35243
Yixuan Wu; Yanzhi Zhang Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization, Discrete and Continuous Dynamical Systems. Series S, Volume 15 (2022) no. 4, pp. 851-876 | DOI:10.3934/dcdss.2022016 | Zbl:1484.47192
Jiří Benedikt; Vladimir Bobkov; Raj Narayan Dhara; Petr Girg Nonradiality of second eigenfunctions of the fractional Laplacian in a ball, Proceedings of the American Mathematical Society, Volume 150 (2022) no. 12, pp. 5335-5348 | DOI:10.1090/proc/16062 | Zbl:1498.35018
Rong Zhang; Xiaoshan Wang; ZuoDong Yang Symmetry and nonexistence of positive solutions for an elliptic system involving the fractional Laplacian, Quaestiones Mathematicae, Volume 45 (2022) no. 2, pp. 247-265 | DOI:10.2989/16073606.2020.1854363 | Zbl:1486.35453
Rongrong Tian; Jinlong Wei; Yanbin Tang The Dirichlet problem for nonlocal elliptic equations, Applicable Analysis, Volume 100 (2021) no. 10, pp. 2093-2107 | DOI:10.1080/00036811.2019.1677893 | Zbl:1472.35422
José C. Bellido; Alejandro Ortega A restricted nonlocal operator bridging together the Laplacian and the fractional Laplacian, Calculus of Variations and Partial Differential Equations, Volume 60 (2021) no. 2, p. 29 (Id/No 71) | DOI:10.1007/s00526-020-01896-1 | Zbl:1462.35431
Dorota Bors; Robert Stańczy Existence and multiplicity results for Dirichlet problem with fractional Laplacian and nonlinearity, Journal of Fixed Point Theory and Applications, Volume 23 (2021) no. 4, p. 16 (Id/No 75) | DOI:10.1007/s11784-021-00915-8 | Zbl:1515.34012
Chunyan Zhang; Jihui Zhang Multiple solutions for the fractional $p$ -Laplacian equation with Hardy-Sobolev exponents, Rocky Mountain Journal of Mathematics, Volume 51 (2021) no. 1, pp. 363-374 | DOI:10.1216/rmj.2021.51.363 | Zbl:1473.35642
Mao-Sheng Chang; Hsi-Chun Wu Existence of weak solutions for the nonlocal energy-weighted fractional reaction-diffusion equations, Journal of Evolution Equations, Volume 19 (2019) no. 3, pp. 883-914 | DOI:10.1007/s00028-019-00494-9 | Zbl:1423.35192
Zu Gao; Xianhua Tang; Sitong Chen Ground state solutions of fractional Choquard equations with general potentials and nonlinearities, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM, Volume 113 (2019) no. 3, pp. 2037-2057 | DOI:10.1007/s13398-018-0598-5 | Zbl:1418.35361
Mahamadi Warma Approximate controllability from the exterior of space-time fractional diffusive equations, SIAM Journal on Control and Optimization, Volume 57 (2019) no. 3, pp. 2037-2063 | DOI:10.1137/18m117145x; | Zbl:1421.93024
Mahamadi Warma Approximate Controllability from the Exterior of Space-Time Fractional Diffusive Equations, SIAM Journal on Control and Optimization, Volume 57 (2019) no. 3, p. 2037 | DOI:10.1137/18m117145x
Anran Li; Chongqing Wei On fractional $p$ -Laplacian problems with local conditions, Advances in Nonlinear Analysis, Volume 7 (2018) no. 4, pp. 485-496 | DOI:10.1515/anona-2016-0105 | Zbl:1404.35480
Gaixia Ning; Zhiyong Wang; Jihui Zhang Existence and multiplicity results for the fractional $p$ -Laplacian equation with Hardy-Sobolev exponents, Differential Equations and Applications, Volume 10 (2018) no. 1, pp. 87-114 | DOI:10.7153/dea-2018-10-06 | Zbl:1391.35399
Huyuan Chen The Dirichlet elliptic problem involving regional fractional Laplacian, Journal of Mathematical Physics, Volume 59 (2018) no. 7, p. 071504 | DOI:10.1063/1.5046685 | Zbl:1394.35549
Pei Ma; Feng Quan Li; Yan Li A Pohozaev identity for the fractional Hénon system, Acta Mathematica Sinica. English Series, Volume 33 (2017) no. 10, pp. 1382-1396 | DOI:10.1007/s10114-017-6556-x | Zbl:1386.35103
Tuhin Ghosh; Yi-Hsuan Lin; Jingni Xiao The Calderón problem for variable coefficients nonlocal elliptic operators, Communications in Partial Differential Equations, Volume 42 (2017) no. 12, pp. 1923-1961 | DOI:10.1080/03605302.2017.1390681 | Zbl:1387.35619
Anran Li; Chongqing Wei Infinitely many solutions for fractional Laplacian problems with local growth conditions, Advances in Difference Equations, Volume 2016 (2016), p. 9 (Id/No 244) | DOI:10.1186/s13662-016-0963-7 | Zbl:1419.35217
Alessio Fiscella; Raffaella Servadei; Enrico Valdinoci Density properties for fractional Sobolev spaces, Annales Academiae Scientiarum Fennicae. Mathematica, Volume 40 (2015) no. 1, pp. 235-253 | Zbl:1346.46025
Xavier Ros-Oton; Joaquim Serra Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Communications in Partial Differential Equations, Volume 40 (2015) no. 1, pp. 115-133 | DOI:10.1080/03605302.2014.918144 | Zbl:1315.35098
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Mahamadi Warma The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Analysis, Volume 42 (2015) no. 2, pp. 499-547 | DOI:10.1007/s11118-014-9443-4 | Zbl:1307.31022
Xiaojun Chang Ground states of some fractional Schrödinger equations on $R^{N}$ , Proceedings of the Edinburgh Mathematical Society. Series II, Volume 58 (2015) no. 2, pp. 305-321 | DOI:10.1017/s0013091514000200 | Zbl:1327.35396
Xavier Ros-Oton; Joaquim Serra The Pohozaev identity for the fractional Laplacian, Archive for Rational Mechanics and Analysis, Volume 213 (2014) no. 2, pp. 587-628 | DOI:10.1007/s00205-014-0740-2 | Zbl:1361.35199
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Roberta Musina; Alexander I. Nazarov On fractional Laplacians, Communications in Partial Differential Equations, Volume 39 (2014) no. 9, pp. 1780-1790 | DOI:10.1080/03605302.2013.864304 | Zbl:1304.47061
Massimiliano Ferrara; Giovanni Molica Bisci; Binlin Zhang Existence of weak solutions for non-local fractional problems via Morse theory, Discrete and Continuous Dynamical Systems. Series B, Volume 19 (2014) no. 8, pp. 2483-2499 | DOI:10.3934/dcdsb.2014.19.2483 | Zbl:1310.35238
Tadeusz Kulczycki; Robert Stańczy Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian, Discrete and Continuous Dynamical Systems. Series B, Volume 19 (2014) no. 8, pp. 2581-2591 | DOI:10.3934/dcdsb.2014.19.2581 | Zbl:1310.35239
Xavier Ros-Oton; Joaquim Serra The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 101 (2014) no. 3, pp. 275-302 | DOI:10.1016/j.matpur.2013.06.003 | Zbl:1285.35020
Dorota Bors Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstract and Applied Analysis, Volume 2013 (2013), p. 10 (Id/No 240863) | DOI:10.1155/2013/240863 | Zbl:1291.45006

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