# American Institute of Mathematical Sciences

November  2018, 17(6): 2547-2575. doi: 10.3934/cpaa.2018121

## Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian

 Universitat Politècnica de Catalunya and BGSMath, Departament de Matemàtiques, Diagonal 647, 08028 Barcelona, Spain

Received  January 2018 Revised  March 2018 Published  June 2018

Fund Project: The author is supported by MINECO grants MDM-2014-0445 and MTM2014-52402-C3-1-P. He is member of the Barcelona Graduate School of Mathematics and part of the Catalan research group 2014 SGR 1083.

We study the regularity of stable solutions to the problem
 \begin{align}\left\{ \begin{gathered} {\left( { - \Delta } \right)^s}&u = f\left( u \right)&{\text{in}}\;\;{B_1}, \hfill \\ &u \equiv 0&{\text{in}}\;\;{{\mathbb{R}}^n}\backslash {B_1}, \hfill \\ \end{gathered} \right.\end{align}
where
 $s∈(0,1)$
. Our main result establishes an
 $L^∞$
bound for stable and radially decreasing
 $H^s$
solutions to this problem in dimensions
 $2 ≤ n < 2(s+2+\sqrt{2(s+1)})$
. In particular, this estimate holds for all
 $s∈(0,1)$
in dimensions
 $2 ≤ n≤ 6$
. It applies to all nonlinearities
 $f∈ C^2$
.
For such parameters
 $s$
and
 $n$
, our result leads to the regularity of the extremal solution when
 $f$
is replaced by
 $λ f$
with
 $λ > 0$
. This is a widely studied question for
 $s = 1$
, which is still largely open in the nonradial case both for
 $s = 1$
and
 $s < 1$
.
Citation: Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
##### References:
 [1] M. Birkner, J. A. López-Mimbela and A. Wakolbinger, Comparison results and steady states for the Fujita equation with fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 83-97.  doi: 10.1016/j.anihpc.2004.05.002. [2] H. Brezis, Is there failure of the inverse function theorem? Morse theory, minimax theory and their applications to nonlinear differential equations, New Stud. Adv. Math., 1 (2003), 23-33. [3] H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469, https://eudml.org/doc/44278. [4] X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380.  doi: 10.1002/cpa.20327. [5] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal., 238 (2006), 709-733.  doi: 10.1016/j.jfa.2005.12.018. [6] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001. [7] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [8] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.  doi: 10.1080/03605302.2011.562954. [9] M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218.  doi: 10.1007/BF00280741. [10] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman and Hall/CRC, 2011. [11] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9. [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer Berlin, New York, 2001. [13] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  doi: 10.1007/BF00250508. [14] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 997-1002.  doi: 10.1016/S0764-4442(00)00289-5. [15] X. Ros-Oton, Regularity for the fractional Gelfand problem up to dimension 7, J. Math. Anal. Appl., 419 (2014), 10-19.  doi: 10.1016/j.jmaa.2014.04.048. [16] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003. [17] X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations, 50 (2014), 723-750.  doi: 10.1007/s00526-013-0653-1. [18] M. Sanchón, Boundedness of the extremal solution of some p-Laplacian problems, Nonlinear Anal., 67 (2007), 281-294.  doi: 10.1016/j.na.2006.05.010. [19] S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.  doi: 10.1016/j.aim.2012.11.015.

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##### References:
 [1] M. Birkner, J. A. López-Mimbela and A. Wakolbinger, Comparison results and steady states for the Fujita equation with fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 83-97.  doi: 10.1016/j.anihpc.2004.05.002. [2] H. Brezis, Is there failure of the inverse function theorem? Morse theory, minimax theory and their applications to nonlinear differential equations, New Stud. Adv. Math., 1 (2003), 23-33. [3] H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469, https://eudml.org/doc/44278. [4] X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380.  doi: 10.1002/cpa.20327. [5] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal., 238 (2006), 709-733.  doi: 10.1016/j.jfa.2005.12.018. [6] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001. [7] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [8] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.  doi: 10.1080/03605302.2011.562954. [9] M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218.  doi: 10.1007/BF00280741. [10] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman and Hall/CRC, 2011. [11] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9. [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer Berlin, New York, 2001. [13] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  doi: 10.1007/BF00250508. [14] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 997-1002.  doi: 10.1016/S0764-4442(00)00289-5. [15] X. Ros-Oton, Regularity for the fractional Gelfand problem up to dimension 7, J. Math. Anal. Appl., 419 (2014), 10-19.  doi: 10.1016/j.jmaa.2014.04.048. [16] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003. [17] X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations, 50 (2014), 723-750.  doi: 10.1007/s00526-013-0653-1. [18] M. Sanchón, Boundedness of the extremal solution of some p-Laplacian problems, Nonlinear Anal., 67 (2007), 281-294.  doi: 10.1016/j.na.2006.05.010. [19] S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.  doi: 10.1016/j.aim.2012.11.015.
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