# American Institute of Mathematical Sciences

January  2011, 10(1): 193-207. doi: 10.3934/cpaa.2011.10.193

## Asymptotic behavior for solutions of some integral equations

 1 School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097, China 2 Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309, United States

Received  January 2010 Revised  June 2010 Published  November 2010

In this paper we study the asymptotic behavior of the positive solutions of the following system of Euler-Lagrange equations of the Hardy-Littlewood-Sobolev type in $R^n$

$u(x) = \frac{1}{|x|^{\alpha}}\int_{R^n} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy$,

$v(x) = \frac{1}{|x|^{\beta}}\int_{R^n} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}}dy.$

We obtain the growth rate of the solutions around the origin and the decay rate near infinity. Some new cases beyond the work of C. Li and J. Lim [17] are studied here. In particular, we remove some technical restrictions of [17], and thus complete the study of the asymptotic behavior of the solutions for non-negative $\alpha$ and $\beta$.

Citation: Yutian Lei, Chao Ma. Asymptotic behavior for solutions of some integral equations. Communications on Pure and Applied Analysis, 2011, 10 (1) : 193-207. doi: 10.3934/cpaa.2011.10.193
##### References:
 [1] 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. doi: doi:10.1002/cpa.3160420304. [2] W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Disc. & Cont. Dynamics Sys. S, (2005), 164-173. [3] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: doi:10.1215/S0012-7094-91-06325-8. [4] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. doi: doi:10.2307/2951844. [5] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure and Appl. Anal., 4 (2005), 1-8. [6] W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962. doi: doi:10.1090/S0002-9939-07-09232-5. [7] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. & Cont. Dynamics Sys., 24 (2009), 1167-1184. [8] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., 59 (2006), 330-343. doi: doi:10.1002/cpa.20116. [9] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. in Partial Differential Equations, 30 (2005), 59-65. doi: doi:10.1081/PDE-200044445. [10] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. & Cont. Dynamics Sys., 12 (2005), 347-354. [11] A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 1-12. [12] L. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems," Cambridge Unversity Press, New York, 2000. [13] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, collected in the book "Mathematical Analysis and Applications," which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981. [14] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: doi:10.1090/S0002-9939-05-08411-X. [15] C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. PDEs, 26 (2006), 447-457. [16] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. [17] C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: doi:10.3934/cpaa.2007.6.453. [18] 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: doi:10.1137/080712301. [19] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: doi:10.2307/2007032. [20] E. Lieb and M. Loss, "Analysis," 2nd edition, American Mathematical Society, Rhode Island, 2001. [21] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 8 (2009), 1925-1932. doi: doi:10.3934/cpaa.2009.8.1925. [22] L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: doi:10.3934/cpaa.2006.5.855. [23] B. Ou, A Remark on a singular integral equation, Houston J. of Math., 25 (1999), 181-184. [24] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. doi: doi:10.1007/BF00250468. [25] E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton University Press, Princeton, 1971. [26] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. [27] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: doi:10.1007/s002080050258.

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##### References:
 [1] 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. doi: doi:10.1002/cpa.3160420304. [2] W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Disc. & Cont. Dynamics Sys. S, (2005), 164-173. [3] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: doi:10.1215/S0012-7094-91-06325-8. [4] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. doi: doi:10.2307/2951844. [5] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure and Appl. Anal., 4 (2005), 1-8. [6] W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962. doi: doi:10.1090/S0002-9939-07-09232-5. [7] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. & Cont. Dynamics Sys., 24 (2009), 1167-1184. [8] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., 59 (2006), 330-343. doi: doi:10.1002/cpa.20116. [9] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. in Partial Differential Equations, 30 (2005), 59-65. doi: doi:10.1081/PDE-200044445. [10] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. & Cont. Dynamics Sys., 12 (2005), 347-354. [11] A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 1-12. [12] L. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems," Cambridge Unversity Press, New York, 2000. [13] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, collected in the book "Mathematical Analysis and Applications," which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981. [14] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: doi:10.1090/S0002-9939-05-08411-X. [15] C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. PDEs, 26 (2006), 447-457. [16] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. [17] C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: doi:10.3934/cpaa.2007.6.453. [18] 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: doi:10.1137/080712301. [19] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: doi:10.2307/2007032. [20] E. Lieb and M. Loss, "Analysis," 2nd edition, American Mathematical Society, Rhode Island, 2001. [21] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 8 (2009), 1925-1932. doi: doi:10.3934/cpaa.2009.8.1925. [22] L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: doi:10.3934/cpaa.2006.5.855. [23] B. Ou, A Remark on a singular integral equation, Houston J. of Math., 25 (1999), 181-184. [24] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. doi: doi:10.1007/BF00250468. [25] E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton University Press, Princeton, 1971. [26] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. [27] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: doi:10.1007/s002080050258.
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