Positive solutions of integral systems involving Bessel potentials

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November 2013, 12(6): 2721-2737. doi: 10.3934/cpaa.2013.12.2721

Positive solutions of integral systems involving Bessel potentials

1.	School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097

Received October 2012 Revised December 2012 Published May 2013

Abstract
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This paper is concerned with integral systems involving the Bessel potentials. Such integral systems are helpful to understand the corresponding PDE systems, such as some static Shrödinger systems with the critical and the supercritical exponents. We use the lifting lemma on regularity to obtain an integrability interval of solutions. Since the Bessel kernel does not have singularity at infinity, we extend the integrability interval to the whole $[1,\infty]$. Next, we use the method of moving planes to prove the radial symmetry for the positive solution of the system. Based on these results, by an iteration we obtain the estimate of the exponential decay of those solutions near infinity. Finally, we discuss the uniqueness of the positive solution of PDE system under some assumption.

Keywords: method of moving planes, integrability intervals, Integral equations, Bessel potential, decay rate., Hardy-Littlewood-Sobolev inequality, radial symmetry.

Mathematics Subject Classification: Primary: 45E10, 45G0.

Citation: Yutian Lei. Positive solutions of integral systems involving Bessel potentials. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2721-2737. doi: 10.3934/cpaa.2013.12.2721

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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