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Remarks on weak solutions of fractional elliptic equations
1. | Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China, China |
References:
[1] |
Ph. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, J. Evolution Eq., 3 (2003), 673-770.
doi: 10.1007/s00028-003-0117-8. |
[2] |
H. Brezis, Some variational problems of the Thomas-Fermi type. Variational inequalities and complementarity problems, Proc. Internat. School, Erice, Wiley, Chichester, (1980), 53-73. |
[3] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[4] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, In Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[5] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, Trans. American Mathematical Society, 367 (2015), 911-941.
doi: 10.1090/S0002-9947-2014-05906-0. |
[6] |
X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732.
doi: 10.1002/cpa.20093. |
[7] |
Z. Chen and R. Song, Estimates on Green functions and poisson kernels for symmetric stable process, Math. Ann., 312 (1998), 465-501.
doi: 10.1007/s002080050232. |
[8] |
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. |
[9] |
H. Chen and L. Véron, Semilinear fractional elliptic equations with gradient nonlinearity involving measures, J. Funct. Anal., 266 (2014), 5467-5492.
doi: 10.1016/j.jfa.2013.11.009. |
[10] |
H. Chen and L. Véron, Weakly and strongly singular solutions of semilinear fractional elliptic equations, Asymptotic Analysis, 88 (2014), 165-184. |
[11] |
H. Chen and J. Yang, Semilinear fractional elliptic equations with measures in unbounded domain, arXiv: 1403.1530. |
[12] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete and Continuous Dynamical Systems, 12 (2005), 347-354. |
[13] |
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. |
[14] |
X. Chen and J. Yang, Limiting behavior of solutions to an equation with the fractional Laplacian, Diff. Integral Equations, 27 (2014), 157-179. |
[15] |
P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian, Comm. Cont. Math., 16 (2014).
doi: 10.1142/S0219199713500235. |
[16] |
M. Marcus and A. C. Ponce, Reduced limits for nonlinear equations with measures, J. Funct. Anal., 258 (2010), 2316-2372.
doi: 10.1016/j.jfa.2009.09.007. |
[17] |
J. Tan, Y. Wang and J. Yang, Nonlinear fractional field equations, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 2098-2110.
doi: 10.1016/j.na.2011.10.010. |
[18] |
L. Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal. T. M. & A., 5 (1981), 225-242.
doi: 10.1016/0362-546X(81)90028-6. |
[19] |
L. Véron, Elliptic equations involving Measures, Stationary Partial Differential Equations, Vol. I, 593-712, Handb. Differ. Eq., North-Holland, Amsterdam, 2004.
doi: 10.1016/S1874-5733(04)80010-X. |
show all references
References:
[1] |
Ph. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, J. Evolution Eq., 3 (2003), 673-770.
doi: 10.1007/s00028-003-0117-8. |
[2] |
H. Brezis, Some variational problems of the Thomas-Fermi type. Variational inequalities and complementarity problems, Proc. Internat. School, Erice, Wiley, Chichester, (1980), 53-73. |
[3] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[4] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, In Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[5] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, Trans. American Mathematical Society, 367 (2015), 911-941.
doi: 10.1090/S0002-9947-2014-05906-0. |
[6] |
X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732.
doi: 10.1002/cpa.20093. |
[7] |
Z. Chen and R. Song, Estimates on Green functions and poisson kernels for symmetric stable process, Math. Ann., 312 (1998), 465-501.
doi: 10.1007/s002080050232. |
[8] |
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. |
[9] |
H. Chen and L. Véron, Semilinear fractional elliptic equations with gradient nonlinearity involving measures, J. Funct. Anal., 266 (2014), 5467-5492.
doi: 10.1016/j.jfa.2013.11.009. |
[10] |
H. Chen and L. Véron, Weakly and strongly singular solutions of semilinear fractional elliptic equations, Asymptotic Analysis, 88 (2014), 165-184. |
[11] |
H. Chen and J. Yang, Semilinear fractional elliptic equations with measures in unbounded domain, arXiv: 1403.1530. |
[12] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete and Continuous Dynamical Systems, 12 (2005), 347-354. |
[13] |
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. |
[14] |
X. Chen and J. Yang, Limiting behavior of solutions to an equation with the fractional Laplacian, Diff. Integral Equations, 27 (2014), 157-179. |
[15] |
P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian, Comm. Cont. Math., 16 (2014).
doi: 10.1142/S0219199713500235. |
[16] |
M. Marcus and A. C. Ponce, Reduced limits for nonlinear equations with measures, J. Funct. Anal., 258 (2010), 2316-2372.
doi: 10.1016/j.jfa.2009.09.007. |
[17] |
J. Tan, Y. Wang and J. Yang, Nonlinear fractional field equations, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 2098-2110.
doi: 10.1016/j.na.2011.10.010. |
[18] |
L. Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal. T. M. & A., 5 (1981), 225-242.
doi: 10.1016/0362-546X(81)90028-6. |
[19] |
L. Véron, Elliptic equations involving Measures, Stationary Partial Differential Equations, Vol. I, 593-712, Handb. Differ. Eq., North-Holland, Amsterdam, 2004.
doi: 10.1016/S1874-5733(04)80010-X. |
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