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Regularity and nonexistence of solutions for a system involving the fractional Laplacian
1. | School of Mathematics, Hunan University, Changsha, 410082, China |
References:
[1] |
J. Betoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. |
[2] |
H. Berestycki and L. Nirenberg, On the method of moving planes and sliding method, Bol. Soc. Brazil. Mat. (N. S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[3] |
G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ and $R_{+}^N$ through the method of moving plane, Comm. PDE., 22 (1997), 1671-1690.
doi: 10.1080/03605309708821315. |
[4] |
K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80. |
[5] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. I. H. Poincaré-AN., 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[6] |
X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), no.12, 1678-1732.
doi: 10.1002/cpa.20093. |
[7] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in PDE., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[8] |
L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Disc. Cont. Dyn. Sys., 33 (2013), 3937-3955.
doi: 10.3934/dcds.2013.33.3937. |
[9] |
W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Mathematics, 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[10] |
W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8. |
[11] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst., 4 2010. |
[12] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[13] |
W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequlaities and systems of integral equations, Discrete Contin. Dyn. Syst., (2005), 164-172. |
[14] |
R. Cont and P. Tankov, Financial Modelling With Jump Process, Chapman and Hall/CRC Financial Mathematics Series, Chapman and Hall/CRC, Boca Raton, FL, 2004. |
[15] |
G. Duvaut and J.-L. Lions, Inequalities In Mechanics and Physics, Springer-Verlag, Berlin, 1976. Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219. |
[16] |
Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.
doi: 10.1016/j.aim.2012.01.018. |
[17] |
P. Felmer and A. Quaas, Fundamental solutions and Liouville type properties for nonlinear integral operator, Adv. Math., 226 (2011), 2712-2738.
doi: 10.1016/j.aim.2010.09.023. |
[18] |
M. Moustapha Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space,, Available online at \emph{http://arxiv.org/abs/1309.7230}., ().
|
[19] |
M. Moustapha Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.
doi: 10.1016/j.jfa.2012.06.018. |
[20] |
R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[21] |
E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312.
doi: 10.1016/j.aim.2007.08.009. |
[22] |
A. Quaas and B. Sirakov, Existence and nonexistence results for fully nonlinear elliptic systems, Indiana Univ. Math. J., 58 (2009), 751-788.
doi: 10.1512/iumj.2009.58.3501. |
[23] |
A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659.
doi: 10.1007/s00526-014-0727-8. |
[24] |
T. Kulczycki, Properties of Green function of symmetric stable processed, Probability and Mathematical Statistics, 17 (1997), 339-364. |
[25] |
L. Silvestre, Regularity of the obstacle problem for the fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[26] |
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. |
show all references
References:
[1] |
J. Betoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. |
[2] |
H. Berestycki and L. Nirenberg, On the method of moving planes and sliding method, Bol. Soc. Brazil. Mat. (N. S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[3] |
G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ and $R_{+}^N$ through the method of moving plane, Comm. PDE., 22 (1997), 1671-1690.
doi: 10.1080/03605309708821315. |
[4] |
K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80. |
[5] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. I. H. Poincaré-AN., 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[6] |
X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), no.12, 1678-1732.
doi: 10.1002/cpa.20093. |
[7] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in PDE., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[8] |
L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Disc. Cont. Dyn. Sys., 33 (2013), 3937-3955.
doi: 10.3934/dcds.2013.33.3937. |
[9] |
W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Mathematics, 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[10] |
W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8. |
[11] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst., 4 2010. |
[12] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[13] |
W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequlaities and systems of integral equations, Discrete Contin. Dyn. Syst., (2005), 164-172. |
[14] |
R. Cont and P. Tankov, Financial Modelling With Jump Process, Chapman and Hall/CRC Financial Mathematics Series, Chapman and Hall/CRC, Boca Raton, FL, 2004. |
[15] |
G. Duvaut and J.-L. Lions, Inequalities In Mechanics and Physics, Springer-Verlag, Berlin, 1976. Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219. |
[16] |
Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.
doi: 10.1016/j.aim.2012.01.018. |
[17] |
P. Felmer and A. Quaas, Fundamental solutions and Liouville type properties for nonlinear integral operator, Adv. Math., 226 (2011), 2712-2738.
doi: 10.1016/j.aim.2010.09.023. |
[18] |
M. Moustapha Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space,, Available online at \emph{http://arxiv.org/abs/1309.7230}., ().
|
[19] |
M. Moustapha Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.
doi: 10.1016/j.jfa.2012.06.018. |
[20] |
R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[21] |
E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312.
doi: 10.1016/j.aim.2007.08.009. |
[22] |
A. Quaas and B. Sirakov, Existence and nonexistence results for fully nonlinear elliptic systems, Indiana Univ. Math. J., 58 (2009), 751-788.
doi: 10.1512/iumj.2009.58.3501. |
[23] |
A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659.
doi: 10.1007/s00526-014-0727-8. |
[24] |
T. Kulczycki, Properties of Green function of symmetric stable processed, Probability and Mathematical Statistics, 17 (1997), 339-364. |
[25] |
L. Silvestre, Regularity of the obstacle problem for the fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[26] |
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. |
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