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

show all references

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

[1]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[2]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274

[3]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[4]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[5]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[6]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[7]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[8]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[9]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[10]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[11]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[12]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[13]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

[14]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[15]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[16]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[17]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[18]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026

[19]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[20]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]