July  2017, 16(4): 1121-1134. doi: 10.3934/cpaa.2017054

Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

2. 

Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China

3. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA

* Corresponding author

Received  September 2014 Revised  June 2016 Published  April 2017

Fund Project: The first author is supported by Fundamental Research Funds for the Central Universities: NS2014080.

The purpose of this paper is to investigate positive solutions of integral equations involving Bessel potential. Exploiting the moving plane method in integral form, we give the radial symmetry of both the domain and solutions of our integral equations in exterior domains and in annular domains respectively.

Citation: Xiaotao Huang, Lihe Wang. Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1121-1134. doi: 10.3934/cpaa.2017054
References:
[1]

R. Adams, Sobolev Spaces, in: Pure Appl. Math. , vol. 65, Academic Press, New York, 1975.

[2]

A. D. Alexandroff, A characteristic property of the spheres, Ann. Math. Pura. Appl., 58 (1962), 303-354.  doi: 10.1007/BF02413056.

[3]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[4]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[5]

L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Tracts in Mathematics, Cmbridge University Press, 2000. doi: 10.1017/CBO9780511569203.

[6]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phy., 68 (1979), 209-243. 

[7]

F. GladialiM. GrossiF. Pacella and P. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var., 40 (2011), 295-317.  doi: 10.1007/s00526-010-0341-3.

[8]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel potential integral equation, J. Diff. Equa., 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.

[9]

X. HuangD. Li and L. Wang, Symmetry of integral equation systems with Bessel kernel on bounded domains, Nonlinear Analysis, 74 (2011), 494-500.  doi: 10.1016/j.na.2010.09.004.

[10]

D. LiG. Ströhmer and L. Wang, Symmetry of integral equations on bounded domains, Proc. Amer. Math. Soc., 137 (2009), 3695-3702.  doi: 10.1090/S0002-9939-09-09987-0.

[11]

Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Diff. Equa., 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T.

[12]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Analysis, 71 (2009), 1796-1906.  doi: 10.1016/j.na.2009.01.014.

[13]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, arXiv: 1101.1649v1. doi: 10.1016/j.na.2011.11.036.

[14]

C. MaW. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020.

[15]

L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2009), 943-949.  doi: 10.1016/j.jmaa.2007.12.064.

[16]

V. Moroz and J. Schaftingen, Nonexistence and optimal decay of supersolutions to choquard equations in exterior domains, J. Diff. Equa., 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.

[17]

W. Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains, Arch. Rational Mech. Anal., 137 (1997), 381-394.  doi: 10.1007/s002050050034.

[18]

W. Reichel, Characterization of balls by Riesz-Potentials, Annali. di. Matematica, 188 (2009), 235-245.  doi: 10.1007/s10231-008-0073-6.

[19]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.

[20]

E. Stein, Singular Integral and Differentiability Properties of Functions, Princeton Ser. Appl. Math. , Vol. 32, Princeton Univ. Press, Princeton, NJ, 1970.

show all references

References:
[1]

R. Adams, Sobolev Spaces, in: Pure Appl. Math. , vol. 65, Academic Press, New York, 1975.

[2]

A. D. Alexandroff, A characteristic property of the spheres, Ann. Math. Pura. Appl., 58 (1962), 303-354.  doi: 10.1007/BF02413056.

[3]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[4]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[5]

L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Tracts in Mathematics, Cmbridge University Press, 2000. doi: 10.1017/CBO9780511569203.

[6]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phy., 68 (1979), 209-243. 

[7]

F. GladialiM. GrossiF. Pacella and P. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var., 40 (2011), 295-317.  doi: 10.1007/s00526-010-0341-3.

[8]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel potential integral equation, J. Diff. Equa., 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.

[9]

X. HuangD. Li and L. Wang, Symmetry of integral equation systems with Bessel kernel on bounded domains, Nonlinear Analysis, 74 (2011), 494-500.  doi: 10.1016/j.na.2010.09.004.

[10]

D. LiG. Ströhmer and L. Wang, Symmetry of integral equations on bounded domains, Proc. Amer. Math. Soc., 137 (2009), 3695-3702.  doi: 10.1090/S0002-9939-09-09987-0.

[11]

Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Diff. Equa., 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T.

[12]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Analysis, 71 (2009), 1796-1906.  doi: 10.1016/j.na.2009.01.014.

[13]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, arXiv: 1101.1649v1. doi: 10.1016/j.na.2011.11.036.

[14]

C. MaW. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020.

[15]

L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2009), 943-949.  doi: 10.1016/j.jmaa.2007.12.064.

[16]

V. Moroz and J. Schaftingen, Nonexistence and optimal decay of supersolutions to choquard equations in exterior domains, J. Diff. Equa., 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.

[17]

W. Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains, Arch. Rational Mech. Anal., 137 (1997), 381-394.  doi: 10.1007/s002050050034.

[18]

W. Reichel, Characterization of balls by Riesz-Potentials, Annali. di. Matematica, 188 (2009), 235-245.  doi: 10.1007/s10231-008-0073-6.

[19]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.

[20]

E. Stein, Singular Integral and Differentiability Properties of Functions, Princeton Ser. Appl. Math. , Vol. 32, Princeton Univ. Press, Princeton, NJ, 1970.

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