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Positive solutions of integral systems involving Bessel potentials
1. | School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097 |
References:
[1] |
J. Bourgain, Global solutions of nonlinear Schrödinger equations, in "Amer. Math. Soc. Colloq. Publ.," 46 AMS, Providence, RI, 1999. |
[2] |
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. |
[3] |
W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. |
[4] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[5] |
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. |
[6] |
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. |
[7] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in "Mathematical Analysis and Applications," vol. 7a Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981. |
[8] |
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. |
[9] |
F. Hang, On the integral systems related to Hardy-Littlewood-sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. |
[10] |
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. |
[11] |
T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 86 (2001), 5043-5046. |
[12] |
Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905.
doi: 10.1007/s00209-012-1036-6. |
[13] |
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. |
[14] |
C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. |
[15] |
C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.
doi: 10.1137/080712301. |
[16] |
Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. |
[17] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. |
[18] |
T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 22 (2005), 403-439.
doi: 10.1016/j.anihpc.2004.03.004. |
[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] |
L. Ma and D. Chen, Radial symmetry and uniqueness for positive solutions of a Schrödinger type system, Math. Comput. Modelling, 49 (2009), 379-385.
doi: 10.1016/j.mcm.2008.06.010. |
[21] |
L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[22] |
J. Smoller, "Shock Waves and Reaction-diffusion Equations," Grundlehren der Mathematischen Wissenschaften, Vol. 258, Springer-Verlag, New York, 1983. |
[23] |
E. Stein, "Singular Integrals and Differentiability Properties of Function," Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970. |
[24] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. |
[25] |
W. Ziemer, "Weakly Differentiable Functions," Graduate Texts in Math. Vol. 120, Springer-Verlag, New York, 1989. |
show all references
References:
[1] |
J. Bourgain, Global solutions of nonlinear Schrödinger equations, in "Amer. Math. Soc. Colloq. Publ.," 46 AMS, Providence, RI, 1999. |
[2] |
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. |
[3] |
W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. |
[4] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[5] |
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. |
[6] |
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. |
[7] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in "Mathematical Analysis and Applications," vol. 7a Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981. |
[8] |
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. |
[9] |
F. Hang, On the integral systems related to Hardy-Littlewood-sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. |
[10] |
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. |
[11] |
T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 86 (2001), 5043-5046. |
[12] |
Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905.
doi: 10.1007/s00209-012-1036-6. |
[13] |
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. |
[14] |
C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. |
[15] |
C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.
doi: 10.1137/080712301. |
[16] |
Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. |
[17] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. |
[18] |
T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 22 (2005), 403-439.
doi: 10.1016/j.anihpc.2004.03.004. |
[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] |
L. Ma and D. Chen, Radial symmetry and uniqueness for positive solutions of a Schrödinger type system, Math. Comput. Modelling, 49 (2009), 379-385.
doi: 10.1016/j.mcm.2008.06.010. |
[21] |
L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[22] |
J. Smoller, "Shock Waves and Reaction-diffusion Equations," Grundlehren der Mathematischen Wissenschaften, Vol. 258, Springer-Verlag, New York, 1983. |
[23] |
E. Stein, "Singular Integrals and Differentiability Properties of Function," Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970. |
[24] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. |
[25] |
W. Ziemer, "Weakly Differentiable Functions," Graduate Texts in Math. Vol. 120, Springer-Verlag, New York, 1989. |
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