On the fractional Lane-Emden equation

Dávila, Juan; Dupaigne, Louis; Wei, Juncheng

doi:10.1090/tran/6872

On the fractional Lane-Emden equation
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by Juan Dávila, Louis Dupaigne and Juncheng Wei PDF

Trans. Amer. Math. Soc. 369 (2017), 6087-6104 Request permission

Abstract:

We classify solutions of finite Morse index of the fractional Lane-Emden equation \[ (-\Delta )^{s} u = \vert u\vert ^{p-1} u\;\; \text {in $\mathbb {R}^n$.} \]

References

Haïm Brezis, Thierry Cazenave, Yvan Martel, and Arthur Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations 1 (1996), no. 1, 73–90. MR 1357955
Haim Brezis and Juan Luis Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469. MR 1605678
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 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
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
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
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
M. Chipot, M. Chlebík, M. Fila, and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\mathbf R^n_{+}$ with a nonlinear boundary condition, J. Math. Anal. Appl. 223 (1998), no. 2, 429–471. MR 1629293, DOI 10.1006/jmaa.1998.5958
J. Dávila, Singular solutions of semi-linear elliptic problems, Handbook of differential equations: stationary partial differential equations. Vol. VI, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 83–176. MR 2569324, DOI 10.1016/S1874-5733(08)80019-8
Juan Dávila, Louis Dupaigne, and Alberto Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal. 261 (2011), no. 1, 218–232. MR 2785899, DOI 10.1016/j.jfa.2010.12.028
Juan Dávila, Louis Dupaigne, and Marcelo Montenegro, The extremal solution of a boundary reaction problem, Commun. Pure Appl. Anal. 7 (2008), no. 4, 795–817. MR 2393398, DOI 10.3934/cpaa.2008.7.795
Juan Dávila, Louis Dupaigne, Kelei Wang, and Juncheng Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math. 258 (2014), 240–285. MR 3190428, DOI 10.1016/j.aim.2014.02.034
Louis Dupaigne, Stable solutions of elliptic partial differential equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 143, Chapman & Hall/CRC, Boca Raton, FL, 2011. MR 2779463, DOI 10.1201/b10802
Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
Mouhamed Moustapha Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential, http://arxiv.org/abs/1109.5530.
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
Alberto Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\Bbb R^N$, J. Math. Pures Appl. (9) 87 (2007), no. 5, 537–561 (English, with English and French summaries). MR 2322150, DOI 10.1016/j.matpur.2007.03.001
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
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883–901. MR 619749, DOI 10.1080/03605308108820196
Junichi Harada, Positive solutions to the Laplace equation with nonlinear boundary conditions on the half space, Calc. Var. Partial Differential Equations 50 (2014), no. 1-2, 399–435. MR 3194688, DOI 10.1007/s00526-013-0640-6
Ira W. Herbst, Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys. 53 (1977), no. 3, 285–294. MR 436854
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241–269. MR 340701, DOI 10.1007/BF00250508
Yan Yan Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 2, 153–180. MR 2055032, DOI 10.4171/jems/6
S. A. Molčanov and E. Ostrovskiĭ, Symmetric stable processes as traces of degenerate diffusion processes. , Teor. Verojatnost. i Primenen. 14 (1969), 127–130 (Russian, with English summary). MR 0247668
Peter Poláčik, Pavol Quittner, and Philippe Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J. 139 (2007), no. 3, 555–579. MR 2350853, DOI 10.1215/S0012-7094-07-13935-8
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
Frank Spitzer, Some theorems concerning $2$-dimensional Brownian motion, Trans. Amer. Math. Soc. 87 (1958), 187–197. MR 104296, DOI 10.1090/S0002-9947-1958-0104296-5
Kelei Wang, Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal. 75 (2012), no. 13, 5238–5260. MR 2927586, DOI 10.1016/j.na.2012.04.041