Symmetry of positive solutions for equations involving higher order fractional Laplacian

Li, Yan; Zhuo, Ran

doi:10.1090/proc/13052

Symmetry of positive solutions for equations involving higher order fractional Laplacian
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by Yan Li and Ran Zhuo PDF

Proc. Amer. Math. Soc. 144 (2016), 4303-4318 Request permission

Abstract:

In this paper, we consider problems associated with the higher order fractional Laplacian. Through the method of moving planes, we derive rotational symmetry of positive solutions and show their dependence on the $x_n$ variable only. We also establish the equivalence between a semilinear higher order fractional Laplacian equation and its corresponding integral equation, so as to further deduce a Liouville type theorem and obtain a priori estimates for positive solutions.

References

David Applebaum, Lévy processes and stochastic calculus, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, Cambridge, 2009. MR 2512800, DOI 10.1017/CBO9780511809781
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, 2nd ed., Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 2001. MR 1805196, DOI 10.1007/978-1-4757-8137-3
A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4) 58 (1962), 303–315. MR 143162, DOI 10.1007/BF02413056
Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR 1406564
Krzysztof Bogdan, Tomasz Grzywny, and MichałRyznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab. 38 (2010), no. 5, 1901–1923. MR 2722789, DOI 10.1214/10-AOP532
K. Bogdan, T. Kulczycki, and Adam Nowak, Gradient estimates for harmonic and $q$-harmonic functions of symmetric stable processes, Illinois J. Math. 46 (2002), no. 2, 541–556. MR 1936936
Matthias Birkner, José Alfredo López-Mimbela, and Anton Wakolbinger, Comparison results and steady states for the Fujita equation with fractional Laplacian, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 1, 83–97 (English, with English and French summaries). MR 2114412, DOI 10.1016/j.anihpc.2004.05.002
H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), no. 1, 1–37. MR 1159383, DOI 10.1007/BF01244896
Jean-Philippe Bouchaud and Antoine Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep. 195 (1990), no. 4-5, 127–293. MR 1081295, DOI 10.1016/0370-1573(90)90099-N
Luis A. Caffarelli and Alexis Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171 (2010), no. 3, 1903–1930. MR 2680400, DOI 10.4007/annals.2010.171.1903
Luis Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245–1260. MR 2354493, DOI 10.1080/03605300600987306
Luis Caffarelli and Luis Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), no. 5, 597–638. MR 2494809, DOI 10.1002/cpa.20274
Linfen Cao and Wenxiong Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst. 33 (2013), no. 9, 3937–3955. MR 3038047, DOI 10.3934/dcds.2013.33.3937
Antonio Capella, Juan Dávila, Louis Dupaigne, and Yannick Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations 36 (2011), no. 8, 1353–1384. MR 2825595, DOI 10.1080/03605302.2011.562954
W. Chen, Y. Fang and Y. Yang, Nonlinear equations involving fractional Laplacians in domains, to appear, Adv. Math.
Zhen-Qing Chen, Panki Kim, and Renming Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1307–1329. MR 2677618, DOI 10.4171/JEMS/231
Wenxiong Chen and Congming Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst. 24 (2009), no. 4, 1167–1184. MR 2505697, DOI 10.3934/dcds.2009.24.1167
Wenxiong Chen and Congming Li, Methods on nonlinear elliptic equations, AIMS Series on Differential Equations & Dynamical Systems, vol. 4, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010. MR 2759774
Wenxiong Chen, Congming Li, and Biao Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst. 12 (2005), no. 2, 347–354. MR 2122171, DOI 10.3934/dcds.2005.12.347
Wenxiong Chen, Congming Li, and Biao Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), no. 3, 330–343. MR 2200258, DOI 10.1002/cpa.20116
Wenxiong Chen, Congming Li, and Biao Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations 30 (2005), no. 1-3, 59–65. MR 2131045, DOI 10.1081/PDE-200044445
W. Chen, C. Li, L. Zhang and T. Cheng, A Liouville theorem for $\alpha$-harmonic functions in $R^n_+$, arXiv:1409.4106v1(2014).
Yonggeun Cho and Tohru Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math. 11 (2009), no. 3, 355–365. MR 2538202, DOI 10.1142/S0219199709003399
Peter Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical foundation of turbulent viscous flows, Lecture Notes in Math., vol. 1871, Springer, Berlin, 2006, pp. 1–43. MR 2196360, DOI 10.1007/11545989_{1}
Rama Cont and Peter Tankov, Financial modelling with jump processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2042661
Xavier Cabré and Yannick Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), no. 1, 23–53. MR 3165278, DOI 10.1016/j.anihpc.2013.02.001
Luis A. Caffarelli, Sandro Salsa, and Luis Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), no. 2, 425–461. MR 2367025, DOI 10.1007/s00222-007-0086-6
Xavier Cabré and Jinggang Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), no. 5, 2052–2093. MR 2646117, DOI 10.1016/j.aim.2010.01.025
Louis Dupaigne and Yannick Sire, A Liouville theorem for non local elliptic equations, Symmetry for elliptic PDEs, Contemp. Math., vol. 528, Amer. Math. Soc., Providence, RI, 2010, pp. 105–114. MR 2759038, DOI 10.1090/conm/528/10417
M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space, Comm. Contemp. Math. doi: 10.1142/S0219199715500121.
Yanqin Fang and Wenxiong Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math. 229 (2012), no. 5, 2835–2867. MR 2889148, DOI 10.1016/j.aim.2012.01.018
M. Fazly and J. Wei, On finite morse index solutions of higher order fractional Lane-Emden equations, arXiv:1410.5400v1(2014).
S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali di Matematica Pura ed Applicata (1923-), doi:10.1007/s10231-014-0462-y.
Tadeusz Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist. 17 (1997), no. 2, Acta Univ. Wratislav. No. 2029, 339–364. MR 1490808
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
Y. Li, A semilinear equation involving the fractional Laplacian in $R^n$, accepted by Acta Mathematica Scientia.
Dongyan Li and Ran Zhuo, An integral equation on half space, Proc. Amer. Math. Soc. 138 (2010), no. 8, 2779–2791. MR 2644892, DOI 10.1090/S0002-9939-10-10368-2
Luciano Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 679–684. MR 803255, DOI 10.1002/cpa.3160380515
Li Ma and Dezhong Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal. 5 (2006), no. 4, 855–859. MR 2246012, DOI 10.3934/cpaa.2006.5.855
Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. MR 2944369, DOI 10.1016/j.bulsci.2011.12.004
Xavier Ros-Oton and Joaquim Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213 (2014), no. 2, 587–628. MR 3211861, DOI 10.1007/s00205-014-0740-2
Xavier Ros-Oton and Joaquim Serra, Local integration by parts and Pohozaev identities for higher order fractional Laplacians, Discrete Contin. Dyn. Syst. 35 (2015), no. 5, 2131–2150. MR 3294242, DOI 10.3934/dcds.2015.35.2131
Xavier Ros-Oton and Joaquim Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), no. 3, 275–302 (English, with English and French summaries). MR 3168912, DOI 10.1016/j.matpur.2013.06.003
Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), no. 1, 67–112. MR 2270163, DOI 10.1002/cpa.20153
Y. Sire and J. C. Wei, On a fractional Henon equation and applications, submitted.
Raffaella Servadei and Enrico Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), no. 2, 887–898. MR 2879266, DOI 10.1016/j.jmaa.2011.12.032
R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, accepted by Disc. Cont. Dyn. Sys.
Ran Zhuo and Dongyan Li, A system of integral equations on half space, J. Math. Anal. Appl. 381 (2011), no. 1, 392–401. MR 2796218, DOI 10.1016/j.jmaa.2011.02.060
Ran Zhuo, Fengquan Li, and Boqiang Lv, Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Commun. Pure Appl. Anal. 13 (2014), no. 3, 977–990. MR 3177684, DOI 10.3934/cpaa.2014.13.977