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 & Applied Analysis, 2017, 16 (4) : 1121-1134. doi: 10.3934/cpaa.2017054
References:
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R. Adams, Sobolev Spaces, in: Pure Appl. Math. , vol. 65, Academic Press, New York, 1975. Google Scholar

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

[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. Google Scholar

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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. Google Scholar

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

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

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[20]

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

show all references

References:
[1]

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

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[20]

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

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