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 & 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. doi: doi:10.1002/cpa.3160420304. Google Scholar [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. Google Scholar [3] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: doi:10.1215/S0012-7094-91-06325-8. Google Scholar [4] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. of Math., 145 (1997), 547. doi: doi:10.2307/2951844. Google Scholar [5] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Comm. Pure and Appl. Anal., 4 (2005), 1. Google Scholar [6] W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality,, Proc. Amer. Math. Soc., 136 (2008), 955. doi: doi:10.1090/S0002-9939-07-09232-5. Google Scholar [7] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. & Cont. Dynamics Sys., 24 (2009), 1167. Google Scholar [8] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl., Math., 59 (2006), 330. doi: doi:10.1002/cpa.20116. Google Scholar [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. doi: doi:10.1081/PDE-200044445. Google Scholar [10] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. & Cont. Dynamics Sys., 12 (2005), 347. Google Scholar [11] A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry,, Math. Res. Letters, 4 (1997), 1. Google Scholar [12] L. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems,", Cambridge Unversity Press, (2000). Google Scholar [13] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, collected in the book, (1981). Google Scholar [14] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: doi:10.1090/S0002-9939-05-08411-X. Google Scholar [15] C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. PDEs, 26 (2006), 447. Google Scholar [16] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221. Google Scholar [17] C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Comm. Pure Appl. Anal., 6 (2007), 453. doi: doi:10.3934/cpaa.2007.6.453. Google Scholar [18] C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, , SIAM J. Math. Anal., 40 (2008), 1049. doi: doi:10.1137/080712301. Google Scholar [19] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349. doi: doi:10.2307/2007032. Google Scholar [20] E. Lieb and M. Loss, "Analysis,", 2nd edition, (2001). Google Scholar [21] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations,, Comm. Pure Appl. Anal., 8 (2009), 1925. doi: doi:10.3934/cpaa.2009.8.1925. Google Scholar [22] L. Ma and D. Chen, A Liouville type theorem for an integral system,, Comm. Pure Appl. Anal., 5 (2006), 855. doi: doi:10.3934/cpaa.2006.5.855. Google Scholar [23] B. Ou, A Remark on a singular integral equation,, Houston J. of Math., 25 (1999), 181. Google Scholar [24] J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. doi: doi:10.1007/BF00250468. Google Scholar [25] E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton University Press, (1971). Google Scholar [26] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. Google Scholar [27] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: doi:10.1007/s002080050258. Google Scholar

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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. doi: doi:10.1002/cpa.3160420304. Google Scholar [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. Google Scholar [3] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: doi:10.1215/S0012-7094-91-06325-8. Google Scholar [4] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. of Math., 145 (1997), 547. doi: doi:10.2307/2951844. Google Scholar [5] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Comm. Pure and Appl. Anal., 4 (2005), 1. Google Scholar [6] W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality,, Proc. Amer. Math. Soc., 136 (2008), 955. doi: doi:10.1090/S0002-9939-07-09232-5. Google Scholar [7] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. & Cont. Dynamics Sys., 24 (2009), 1167. Google Scholar [8] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl., Math., 59 (2006), 330. doi: doi:10.1002/cpa.20116. Google Scholar [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. doi: doi:10.1081/PDE-200044445. Google Scholar [10] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. & Cont. Dynamics Sys., 12 (2005), 347. Google Scholar [11] A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry,, Math. Res. Letters, 4 (1997), 1. Google Scholar [12] L. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems,", Cambridge Unversity Press, (2000). Google Scholar [13] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, collected in the book, (1981). Google Scholar [14] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: doi:10.1090/S0002-9939-05-08411-X. Google Scholar [15] C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. PDEs, 26 (2006), 447. Google Scholar [16] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221. Google Scholar [17] C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Comm. Pure Appl. Anal., 6 (2007), 453. doi: doi:10.3934/cpaa.2007.6.453. Google Scholar [18] C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, , SIAM J. Math. Anal., 40 (2008), 1049. doi: doi:10.1137/080712301. Google Scholar [19] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349. doi: doi:10.2307/2007032. Google Scholar [20] E. Lieb and M. Loss, "Analysis,", 2nd edition, (2001). Google Scholar [21] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations,, Comm. Pure Appl. Anal., 8 (2009), 1925. doi: doi:10.3934/cpaa.2009.8.1925. Google Scholar [22] L. Ma and D. Chen, A Liouville type theorem for an integral system,, Comm. Pure Appl. Anal., 5 (2006), 855. doi: doi:10.3934/cpaa.2006.5.855. Google Scholar [23] B. Ou, A Remark on a singular integral equation,, Houston J. of Math., 25 (1999), 181. Google Scholar [24] J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. doi: doi:10.1007/BF00250468. Google Scholar [25] E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton University Press, (1971). Google Scholar [26] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. Google Scholar [27] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: doi:10.1007/s002080050258. Google Scholar
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