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Qualitative properties of solutions to an integral system associated with the Bessel potential
1. | School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China, China |
2. | Department of Mathematics, Wayne State University, Detroit, MI 48202 |
References:
[1] |
J. Bao, N. Lam and G. Lu, Polyharmonic equations with critical exponential growth in the whole space $\mathbb{R}^{N}$, Discrete Contin. Dyn. Syst., 36 (2016), 577-600. |
[2] |
L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[3] |
A. Chang and P. Yang, On uniqueness of solutions of nth order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102.
doi: 10.4310/MRL.1997.v4.n1.a9. |
[4] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys., Vol. 4, 2010. |
[5] |
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. |
[6] |
W. Chen and C. Li, Regularity of solutions for a system of integral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. |
[7] |
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. |
[8] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65. |
[9] |
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. |
[10] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. |
[11] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Mathematical Analysis and Applications, vol. 7a of the book series Advances in Mathematics, Academic Press, New York, 1981. |
[12] |
X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potential, Commun. Pure Appl. Anal., 10 (2011), 1111-1119.
doi: 10.3934/cpaa.2011.10.1111. |
[13] |
X. Han, G. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602. |
[14] |
C. Jin and C. Li, Symmetry of solution to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
doi: 10.1090/S0002-9939-05-08411-X. |
[15] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[16] |
N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth, Discrete Contin. Dyn. Syst., 32 (2012), 2187-2205.
doi: 10.3934/dcds.2012.32.2187. |
[17] |
C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Commun. Pure Appl. Anal., 6 (2007), 453-464.
doi: 10.3934/cpaa.2007.6.453. |
[18] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699. |
[19] |
L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2008), 943-949.
doi: 10.1016/j.jmaa.2007.12.064. |
[20] |
W. Reichel, Characterization of balls by Riesz-potentials, Ann. Mat., 188 (2009), 235-245.
doi: 10.1007/s10231-008-0073-6. |
[21] |
J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318. |
[22] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer-Verlag, New York, Berlin, 1983. |
[23] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Ser. Appl. Math., vol. 32, Princeton Univ. Press, Princeton, NJ, 1970. |
show all references
References:
[1] |
J. Bao, N. Lam and G. Lu, Polyharmonic equations with critical exponential growth in the whole space $\mathbb{R}^{N}$, Discrete Contin. Dyn. Syst., 36 (2016), 577-600. |
[2] |
L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[3] |
A. Chang and P. Yang, On uniqueness of solutions of nth order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102.
doi: 10.4310/MRL.1997.v4.n1.a9. |
[4] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys., Vol. 4, 2010. |
[5] |
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. |
[6] |
W. Chen and C. Li, Regularity of solutions for a system of integral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. |
[7] |
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. |
[8] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65. |
[9] |
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. |
[10] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. |
[11] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Mathematical Analysis and Applications, vol. 7a of the book series Advances in Mathematics, Academic Press, New York, 1981. |
[12] |
X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potential, Commun. Pure Appl. Anal., 10 (2011), 1111-1119.
doi: 10.3934/cpaa.2011.10.1111. |
[13] |
X. Han, G. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602. |
[14] |
C. Jin and C. Li, Symmetry of solution to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
doi: 10.1090/S0002-9939-05-08411-X. |
[15] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[16] |
N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth, Discrete Contin. Dyn. Syst., 32 (2012), 2187-2205.
doi: 10.3934/dcds.2012.32.2187. |
[17] |
C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Commun. Pure Appl. Anal., 6 (2007), 453-464.
doi: 10.3934/cpaa.2007.6.453. |
[18] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699. |
[19] |
L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2008), 943-949.
doi: 10.1016/j.jmaa.2007.12.064. |
[20] |
W. Reichel, Characterization of balls by Riesz-potentials, Ann. Mat., 188 (2009), 235-245.
doi: 10.1007/s10231-008-0073-6. |
[21] |
J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318. |
[22] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer-Verlag, New York, Berlin, 1983. |
[23] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Ser. Appl. Math., vol. 32, Princeton Univ. Press, Princeton, NJ, 1970. |
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