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Two problems related to prescribed curvature measures
Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality
1. | Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097, China |
2. | School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116, China |
References:
[1] |
L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[2] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.
doi: 10.1007/s00032-008-0090-3. |
[3] |
A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 91-102. |
[4] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[5] |
W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564.
doi: 10.2307/2951844. |
[6] |
W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962.
doi: 10.1090/S0002-9939-07-09232-5. |
[7] |
W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, arXiv:1110.2539v1, 2011. |
[8] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[9] |
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. |
[10] |
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. |
[11] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Collected in the book Mathematical\, Analysis and Applications, Which is 7a of the Book Series Advances in Mathematics. Supplementary Studies, Academic Press, New York, (1981). |
[12] |
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 1977. |
[13] |
C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
doi: 10.1090/S0002-9939-05-08411-X. |
[14] |
C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[15] |
Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation, Discrete Contin. Dyn. Syst., 30 (2011), 547-558.
doi: 10.3934/dcds.2011.30.547. |
[16] |
Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61.
doi: 10.1007/s00526-011-0450-7. |
[17] |
Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations, Comm. Pure Appl. Anal., 10 (2011), 193-207.
doi: 10.3934/cpaa.2011.10.193. |
[18] |
C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
doi: 10.1007/s002220050023. |
[19] |
C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464.
doi: 10.3934/cpaa.2007.6.453. |
[20] |
Y. Li, Remark on some conformally invariant integral equations: The method of moving planes, J. Eur. Math. Soc., 6 (2004), 153-180. |
[21] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[22] |
J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$, J. Differential equations, 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[23] |
M. Onodera, On the shape of solutions to an integral system related to the weighted Hardy-Littlewood-Sobolev inequality, J. Math. Anal. Appl., 389 (2012), 498-510.
doi: 10.1016/j.jmaa.2011.12.004. |
[24] |
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. |
[25] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Function," Princetion Math. Series, 30. Princetion University Press, (1970), xiv+290 pp. |
[26] |
E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. |
[27] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: 10.1007/s002080050258. |
show all references
References:
[1] |
L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[2] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.
doi: 10.1007/s00032-008-0090-3. |
[3] |
A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 91-102. |
[4] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[5] |
W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564.
doi: 10.2307/2951844. |
[6] |
W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962.
doi: 10.1090/S0002-9939-07-09232-5. |
[7] |
W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, arXiv:1110.2539v1, 2011. |
[8] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[9] |
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. |
[10] |
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. |
[11] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Collected in the book Mathematical\, Analysis and Applications, Which is 7a of the Book Series Advances in Mathematics. Supplementary Studies, Academic Press, New York, (1981). |
[12] |
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 1977. |
[13] |
C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
doi: 10.1090/S0002-9939-05-08411-X. |
[14] |
C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[15] |
Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation, Discrete Contin. Dyn. Syst., 30 (2011), 547-558.
doi: 10.3934/dcds.2011.30.547. |
[16] |
Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61.
doi: 10.1007/s00526-011-0450-7. |
[17] |
Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations, Comm. Pure Appl. Anal., 10 (2011), 193-207.
doi: 10.3934/cpaa.2011.10.193. |
[18] |
C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
doi: 10.1007/s002220050023. |
[19] |
C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464.
doi: 10.3934/cpaa.2007.6.453. |
[20] |
Y. Li, Remark on some conformally invariant integral equations: The method of moving planes, J. Eur. Math. Soc., 6 (2004), 153-180. |
[21] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[22] |
J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^n$, J. Differential equations, 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[23] |
M. Onodera, On the shape of solutions to an integral system related to the weighted Hardy-Littlewood-Sobolev inequality, J. Math. Anal. Appl., 389 (2012), 498-510.
doi: 10.1016/j.jmaa.2011.12.004. |
[24] |
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. |
[25] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Function," Princetion Math. Series, 30. Princetion University Press, (1970), xiv+290 pp. |
[26] |
E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. |
[27] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: 10.1007/s002080050258. |
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