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January  2018, 17(1): 85-112. doi: 10.3934/cpaa.2018006

Liouville results for fully nonlinear integral elliptic equations in exterior domains

Hongxia Zhang and  Ying Wang ,

Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

* Corresponding author

Received  January 2017 Revised  June 2017 Published  September 2017

Fund Project: Y. Wang is supported by NSFC, No: 11661045.

In this paper, we obtain Liouville type theorems both in the whole space and exterior domain in viscosity sense for fully nonlinear elliptic inequality involving nonlocal Pucci's operator. The nonlocal property of the operator, we only have a much weaker comparison principle, compared with the inequality with classical Pucci's operators, which give rise to the difficulties for the Hadamard type property in exterior domain.

Keywords: Fully nonlinear equation, nonlocal operator, Liouville theorem.
Mathematics Subject Classification: Primary:35J75;Secondary:35B40.
Citation: Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure and Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006
References:
[1]

S. Alarcón, J. García-Melián and A. Quaas, Liouville type theorems for elliptic equations with gradient terms, Milan Journal of Mathematics, 81 (2013), 171-185.  doi: 10.1007/s00032-013-0197-z.

[2]

S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations, Comm. Pure Appl. Math., 64 (2011), 737-777.  doi: 10.1002/cpa.20360.

[3]

S. N. Armstrong and B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Diff. Eq., 36 (2011), 2011-2047.  doi: 10.1080/03605302.2010.534523.

[4]

S. N. Armstrong and B. Sirakov, Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities, Ann. della Scuola Normale Super. di Pisa. Classe di scienze, 10 (2011), 711-728. 

[5]

M. Bardi, A. Cesaroni and L. Rossi, Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control, ESAIM: Control, Optimisation and Calculus of Variations, 22 (2016), 842-861.  doi: 10.1051/cocv/2015033.

[6]

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.

[7]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.  doi: 10.1002/cpa.20274.

[8]

I. Capuzzo-Dolcetta and A. Cutrí, Hadamard and Liouville type results for fully nonlinear partial differential inequalities, Comm. Contemp. Math., 5 (2003), 435-448.  doi: 10.1142/S0219199703001014.

[9]

H. Chen and P. Felmer, On the Liouville Property for fully nonlinear elliptic equations with gradient term, J. Diff. Eq., 255 (2013), 2167-2195.  doi: 10.1016/j.jde.2013.06.009.

[10]

H. Chen, P. Felmer and A. Quaas, Large solution to elliptic equations involving fractional Laplacian, Annales de l'Institut Henri Poincaré, 32 (2015), 1199-1228.  doi: 10.1016/j.anihpc.2014.08.001.

[11]

H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures, J. Diff. Eq., 257 (2014), 1457-1486.  doi: 10.1016/j.jde.2014.05.012.

[12]

W. Chen, Y. Fang and Y. Ray, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[13]

W. Chen and Y. Fang, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Adv. Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.

[14]

W. Chen, X. Cui, Z. Yuan and R. Zhuo, A liouville theorem for the fractional laplacian, arXiv: 1401.7402 (2014). doi: 10.1016/j.na.2014.11.003.

[15]

A. Cutrí and F. Leoni, On the Liouville Property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 219-245.  doi: 10.1016/S0294-1449(00)00109-8.

[16]

M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half-space, Comm. Cont. Math., 18 (2016), 1-25.  doi: 10.1142/S0219199715500121.

[17]

A. Farina and E. Valdinoci, Regularity and rigidity theorems for a class of anisotropic nonlocal operators, Manuscripta Math., (2016).  doi: 10.1007/s00229-016-0875-6.

[18]

P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian, Comm. Contem. Math., 16 (2013).  doi: 10.1142/S0219199713500235.

[19]

P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math., 226 (2011), 2712-2738.  doi: 10.1016/j.aim.2010.09.023.

[20]

P. Felmer and A. Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (2009), 5721-5736.  doi: 10.1090/S0002-9947-09-04566-8.

[21]

P. Felmer and A. Quaas, Fundamental solutions for a class of Isaacs integral operators, Disc. Cont. Dyn. Sys., 30 (2011), 493-508.  doi: 10.3934/dcds.2011.30.493.

[22]

H. Hajaiej, Variational problems related to some fractional kinetic equations, (2012), arXiv: 1205.1202.

[23]

H. Hajaiej, Existence of minimizers of functionals involving the fractional gradient in the abscence of compactness, symmetry and monotonicity, J. Math. Anal. Appl., 399 (2013), 17-26.  doi: 10.1016/j.jmaa.2012.09.023.

[24]

R. Servadei and E. Valdinoci, Moutain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.

[25]

Y. Sire and E. Valdinoci, Fractional laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.

[26]

X. Ros-Oton and J. Serra, Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Commun. Part. Diff. Eq., 40 (2015), 115-133.  doi: 10.1080/03605302.2014.918144.

[27]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

show all references

References:
[1]

S. Alarcón, J. García-Melián and A. Quaas, Liouville type theorems for elliptic equations with gradient terms, Milan Journal of Mathematics, 81 (2013), 171-185.  doi: 10.1007/s00032-013-0197-z.

[2]

S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations, Comm. Pure Appl. Math., 64 (2011), 737-777.  doi: 10.1002/cpa.20360.

[3]

S. N. Armstrong and B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Diff. Eq., 36 (2011), 2011-2047.  doi: 10.1080/03605302.2010.534523.

[4]

S. N. Armstrong and B. Sirakov, Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities, Ann. della Scuola Normale Super. di Pisa. Classe di scienze, 10 (2011), 711-728. 

[5]

M. Bardi, A. Cesaroni and L. Rossi, Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control, ESAIM: Control, Optimisation and Calculus of Variations, 22 (2016), 842-861.  doi: 10.1051/cocv/2015033.

[6]

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.

[7]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.  doi: 10.1002/cpa.20274.

[8]

I. Capuzzo-Dolcetta and A. Cutrí, Hadamard and Liouville type results for fully nonlinear partial differential inequalities, Comm. Contemp. Math., 5 (2003), 435-448.  doi: 10.1142/S0219199703001014.

[9]

H. Chen and P. Felmer, On the Liouville Property for fully nonlinear elliptic equations with gradient term, J. Diff. Eq., 255 (2013), 2167-2195.  doi: 10.1016/j.jde.2013.06.009.

[10]

H. Chen, P. Felmer and A. Quaas, Large solution to elliptic equations involving fractional Laplacian, Annales de l'Institut Henri Poincaré, 32 (2015), 1199-1228.  doi: 10.1016/j.anihpc.2014.08.001.

[11]

H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures, J. Diff. Eq., 257 (2014), 1457-1486.  doi: 10.1016/j.jde.2014.05.012.

[12]

W. Chen, Y. Fang and Y. Ray, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[13]

W. Chen and Y. Fang, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Adv. Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.

[14]

W. Chen, X. Cui, Z. Yuan and R. Zhuo, A liouville theorem for the fractional laplacian, arXiv: 1401.7402 (2014). doi: 10.1016/j.na.2014.11.003.

[15]

A. Cutrí and F. Leoni, On the Liouville Property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 219-245.  doi: 10.1016/S0294-1449(00)00109-8.

[16]

M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half-space, Comm. Cont. Math., 18 (2016), 1-25.  doi: 10.1142/S0219199715500121.

[17]

A. Farina and E. Valdinoci, Regularity and rigidity theorems for a class of anisotropic nonlocal operators, Manuscripta Math., (2016).  doi: 10.1007/s00229-016-0875-6.

[18]

P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian, Comm. Contem. Math., 16 (2013).  doi: 10.1142/S0219199713500235.

[19]

P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math., 226 (2011), 2712-2738.  doi: 10.1016/j.aim.2010.09.023.

[20]

P. Felmer and A. Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (2009), 5721-5736.  doi: 10.1090/S0002-9947-09-04566-8.

[21]

P. Felmer and A. Quaas, Fundamental solutions for a class of Isaacs integral operators, Disc. Cont. Dyn. Sys., 30 (2011), 493-508.  doi: 10.3934/dcds.2011.30.493.

[22]

H. Hajaiej, Variational problems related to some fractional kinetic equations, (2012), arXiv: 1205.1202.

[23]

H. Hajaiej, Existence of minimizers of functionals involving the fractional gradient in the abscence of compactness, symmetry and monotonicity, J. Math. Anal. Appl., 399 (2013), 17-26.  doi: 10.1016/j.jmaa.2012.09.023.

[24]

R. Servadei and E. Valdinoci, Moutain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.

[25]

Y. Sire and E. Valdinoci, Fractional laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.

[26]

X. Ros-Oton and J. Serra, Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Commun. Part. Diff. Eq., 40 (2015), 115-133.  doi: 10.1080/03605302.2014.918144.

[27]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

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