Exact asymptotic behavior of singular positive solutions of fractional semi-linear elliptic equations

Yang, Hui; Zou, Wenming

doi:10.1090/proc/14448

Exact asymptotic behavior of singular positive solutions of fractional semi-linear elliptic equations
HTML articles powered by AMS MathViewer

by Hui Yang and Wenming Zou PDF

Proc. Amer. Math. Soc. 147 (2019), 2999-3009 Request permission

Abstract:

In this paper, we prove the exact asymptotic behavior of singular positive solutions of fractional semi-linear equations \begin{equation*} (-\Delta )^\sigma u = u^p \textrm { in } B_1\backslash \{0\} \end{equation*} with an isolated singularity, where $\sigma \in (0, 1)$ and $\frac {n}{n-2\sigma } < p < \frac {n+2\sigma }{n-2\sigma }$.

References

W. Ao, H. Chan, A. DelaTorre, M.A. Fontelos, M. González, and J. Wei, On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program, arXiv:1802.07973, 2018.
Patricio Aviles, Local behavior of solutions of some elliptic equations, Comm. Math. Phys. 108 (1987), no. 2, 177–192. MR 875297
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
Marie-Françoise Bidaut-Véron and Laurent Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), no. 3, 489–539. MR 1134481, DOI 10.1007/BF01243922
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, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, DOI 10.1002/cpa.3160420304
Luis Caffarelli, Tianling Jin, Yannick Sire, and Jingang Xiong, Local analysis of solutions of fractional semi-linear elliptic equations with isolated singularities, Arch. Ration. Mech. Anal. 213 (2014), no. 1, 245–268. MR 3198648, DOI 10.1007/s00205-014-0722-4
L. Caffarelli, J.-M. Roquejoffre, and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144. MR 2675483, DOI 10.1002/cpa.20331
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
Sun-Yung Alice Chang and María del Mar González, Fractional Laplacian in conformal geometry, Adv. Math. 226 (2011), no. 2, 1410–1432. MR 2737789, DOI 10.1016/j.aim.2010.07.016
Chiun-Chuan Chen and Chang-Shou Lin, On the asymptotic symmetry of singular solutions of the scalar curvature equations, Math. Ann. 313 (1999), no. 2, 229–245. MR 1679784, DOI 10.1007/s002080050259
Huyuan Chen and Alexander Quaas, Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results, J. Lond. Math. Soc. (2) 97 (2018), no. 2, 196–221. MR 3789844, DOI 10.1112/jlms.12104
Wenxiong Chen, Congming Li, and Yan Li, A direct method of moving planes for the fractional Laplacian, Adv. Math. 308 (2017), 404–437. MR 3600062, DOI 10.1016/j.aim.2016.11.038
Juan Dávila, Louis Dupaigne, and Juncheng Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6087–6104. MR 3660214, DOI 10.1090/tran/6872
M. Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential, arXiv:1109.5530, 2011.
Mouhamed Moustapha Fall and Veronica Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations 39 (2014), no. 2, 354–397. MR 3169789, DOI 10.1080/03605302.2013.825918
Rupert L. Frank, Enno Lenzmann, and Luis Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016), no. 9, 1671–1726. MR 3530361, DOI 10.1002/cpa.21591
M. Ghergu, S. Kim, and H. Shahgholian, Isolated singularities for semilinear elliptic systems with power-law nonlinearity, arXiv:1804.04291, 2018.
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. MR 615628, DOI 10.1002/cpa.3160340406
Maria del Mar González, Rafe Mazzeo, and Yannick Sire, Singular solutions of fractional order conformal Laplacians, J. Geom. Anal. 22 (2012), no. 3, 845–863. MR 2927681, DOI 10.1007/s12220-011-9217-9
María del Mar González and Jie Qing, Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE 6 (2013), no. 7, 1535–1576. MR 3148060, DOI 10.2140/apde.2013.6.1535
Qinian Jin, YanYan Li, and Haoyuan Xu, Symmetry and asymmetry: the method of moving spheres, Adv. Differential Equations 13 (2008), no. 7-8, 601–640. MR 2479025
Tianling Jin, Olivaine S. de Queiroz, Yannick Sire, and Jingang Xiong, On local behavior of singular positive solutions to nonlocal elliptic equations, Calc. Var. Partial Differential Equations 56 (2017), no. 1, Paper No. 9, 25. MR 3592764, DOI 10.1007/s00526-016-1102-8
Tianling Jin, YanYan Li, and Jingang Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 6, 1111–1171. MR 3226738, DOI 10.4171/JEMS/456
Nick Korevaar, Rafe Mazzeo, Frank Pacard, and Richard Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math. 135 (1999), no. 2, 233–272. MR 1666838, DOI 10.1007/s002220050285
YanYan Li and Lei Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 (2003), 27–87. MR 2001065, DOI 10.1007/BF02786551
Yanyan Li and Meijun Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), no. 2, 383–417. MR 1369398, DOI 10.1215/S0012-7094-95-08016-8
P.-L. Lions, Isolated singularities in semilinear problems, J. Differential Equations 38 (1980), no. 3, 441–450. MR 605060, DOI 10.1016/0022-0396(80)90018-2
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
H. Yang and W. Zou, On isolated singularities of fractional semi-linear elliptic equations, arXiv:1804.00817, 2018.