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Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations
Maximum principles for a fully nonlinear nonlocal equation on unbounded domains
1. | School of Science, Minzu University of China, Beijing 100081, China |
2. | Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China |
$ \mathcal {F}_{\alpha}(u(x)) = C_{n, \alpha}P.V.\int_{ \mathbb R^n}\frac{G(u(x)-u(z))}{|x-z|^{n+\alpha}}dz = f(u(x)), \; \; \; x\in \mathbb R^n. $ |
References:
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F. Andreu, J. Mazon, J. Rossi and J. Toledo, Nonlocal Diffusion Problems, Vol. 165, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/surv/165. |
[2] |
D. Applebaum,
Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
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[3] |
C. Bjorland, L. Caffarelli and A. Figalli,
Nonlocal tug-of-war and the infinity fractional Laplacian, Commun. Pure Appl. Math., 65 (2012), 337-380.
doi: 10.1002/cpa.21379. |
[4] |
K. Bogdan, T. Kulczycki and A. Nowak,
gradient estimates for harmonic and $q$-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556.
|
[5] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Commun. Patial Differ. Equ., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[6] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-diffenrential equations, Commun. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[7] |
W. Chen, L. D. D'Ambrosio and Y. Li,
Some Liouville theorems for the fractional Laplacian, Nonlinear Anal. Theory Meth. Appl., 121 (2015), 370-381.
doi: 10.1016/j.na.2014.11.003. |
[8] |
W. Chen and C. Li,
Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.
doi: 10.1016/j.aim.2018.07.016. |
[9] |
W. Chen, C. Li and G. Li, Maximum priciple for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differ. Equ., 56 (2017), 29 pp.
doi: 10.1007/s00526-017-1110-3. |
[10] |
W. Chen, C. Li and Y. Li,
A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[11] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[12] |
W. Chen, C. Li and B. Ou,
Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
doi: 10.3934/dcds.2005.12.347. |
[13] |
W. Chen and L. Wu, A maximum principle on unbounded domains and a Liouville theorem for fractional $p$-harmonic functions, preprint, arXiv: 1905.09986. |
[14] |
W. Chen and J. Zhu,
Indefinite fractional elliptic problem and Liouville theorems, J. Differ. Equ., 260 (2016), 4758-4785.
doi: 10.1016/j.jde.2015.11.029. |
[15] |
T. Cheng,
Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599.
doi: 10.3934/dcds.2017154. |
[16] |
T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math., 19 (2017), Art. 1750018.
doi: 10.1142/S0219199717500183. |
[17] |
E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, Pitagora, (1979), 131–188. |
[18] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ., 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[19] |
X. Han, G. Lu and J. Zhu,
Characterization of balls in terms of Bessel-potential integral equation, J. Differ. Equ., 252 (2012), 1589-1602.
doi: 10.1016/j.jde.2011.07.037. |
[20] |
C. Kenig and W. Ni,
An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math., 106 (1984), 689-702.
doi: 10.2307/2374291. |
[21] |
L. Ma and D. Chen,
A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.
doi: 10.3934/cpaa.2006.5.855. |
[22] |
P. Polacik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part Ⅰ: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[23] |
P. Wang and P. Niu,
A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pure Appl. Anal., 16 (2017), 1707-1718.
doi: 10.3934/cpaa.2017082. |
[24] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and nonexistence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
show all references
References:
[1] |
F. Andreu, J. Mazon, J. Rossi and J. Toledo, Nonlocal Diffusion Problems, Vol. 165, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/surv/165. |
[2] |
D. Applebaum,
Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
|
[3] |
C. Bjorland, L. Caffarelli and A. Figalli,
Nonlocal tug-of-war and the infinity fractional Laplacian, Commun. Pure Appl. Math., 65 (2012), 337-380.
doi: 10.1002/cpa.21379. |
[4] |
K. Bogdan, T. Kulczycki and A. Nowak,
gradient estimates for harmonic and $q$-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556.
|
[5] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Commun. Patial Differ. Equ., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[6] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-diffenrential equations, Commun. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[7] |
W. Chen, L. D. D'Ambrosio and Y. Li,
Some Liouville theorems for the fractional Laplacian, Nonlinear Anal. Theory Meth. Appl., 121 (2015), 370-381.
doi: 10.1016/j.na.2014.11.003. |
[8] |
W. Chen and C. Li,
Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.
doi: 10.1016/j.aim.2018.07.016. |
[9] |
W. Chen, C. Li and G. Li, Maximum priciple for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differ. Equ., 56 (2017), 29 pp.
doi: 10.1007/s00526-017-1110-3. |
[10] |
W. Chen, C. Li and Y. Li,
A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[11] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[12] |
W. Chen, C. Li and B. Ou,
Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
doi: 10.3934/dcds.2005.12.347. |
[13] |
W. Chen and L. Wu, A maximum principle on unbounded domains and a Liouville theorem for fractional $p$-harmonic functions, preprint, arXiv: 1905.09986. |
[14] |
W. Chen and J. Zhu,
Indefinite fractional elliptic problem and Liouville theorems, J. Differ. Equ., 260 (2016), 4758-4785.
doi: 10.1016/j.jde.2015.11.029. |
[15] |
T. Cheng,
Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599.
doi: 10.3934/dcds.2017154. |
[16] |
T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math., 19 (2017), Art. 1750018.
doi: 10.1142/S0219199717500183. |
[17] |
E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, Pitagora, (1979), 131–188. |
[18] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ., 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[19] |
X. Han, G. Lu and J. Zhu,
Characterization of balls in terms of Bessel-potential integral equation, J. Differ. Equ., 252 (2012), 1589-1602.
doi: 10.1016/j.jde.2011.07.037. |
[20] |
C. Kenig and W. Ni,
An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math., 106 (1984), 689-702.
doi: 10.2307/2374291. |
[21] |
L. Ma and D. Chen,
A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.
doi: 10.3934/cpaa.2006.5.855. |
[22] |
P. Polacik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part Ⅰ: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[23] |
P. Wang and P. Niu,
A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pure Appl. Anal., 16 (2017), 1707-1718.
doi: 10.3934/cpaa.2017082. |
[24] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and nonexistence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
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