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The exact asymptotic behaviour of the unique solution to a singular Dirichlet problem

Boundary Value Problems volume 2006, Article number: 75674 (2006) Cite this article

Abstract

By Karamata regular variation theory, we show the existence and exact asymptotic behaviour of the unique classical solution near the boundary to a singular Dirichlet problem,,,, where is a bounded domain with smooth boundary in,,, for each and some; and for some, which is nonnegative on and may be unbounded or singular on the boundary.

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References

  1. Crandall MG, Rabinowitz PH, Tartar L: On a Dirichlet problem with a singular nonlinearity. Communications in Partial Differential Equations 1977,2(2):193–222. 10.1080/03605307708820029

    MathSciNet  Article  MATH  Google Scholar 

  2. Fulks W, Maybee JS: A singular non-linear equation. Osaka Journal of Mathematics 1960, 12: 1–19.

    MathSciNet  MATH  Google Scholar 

  3. Nachman A, Callegari A: A nonlinear singular boundary value problem in the theory of pseudoplastic fluids. SIAM Journal on Applied Mathematics 1980,38(2):275–281. 10.1137/0138024

    MathSciNet  Article  MATH  Google Scholar 

  4. Stuart CA: Existence and approximation of solutions of non-linear elliptic equations. Mathematische Zeitschrift 1976,147(1):53–63. 10.1007/BF01214274

    MathSciNet  Article  MATH  Google Scholar 

  5. Díaz G, Letelier R: Explosive solutions of quasilinear elliptic equations: existence and uniqueness. Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1993,20(2):97–125.

    Article  MathSciNet  MATH  Google Scholar 

  6. Lasry J-M, Lions P-L: Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem. Mathematische Annalen 1989,283(4):583–630. 10.1007/BF01442856

    MathSciNet  Article  MATH  Google Scholar 

  7. Cui S: Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems. Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2000,41(1–2):149–176.

    Article  MathSciNet  MATH  Google Scholar 

  8. Ghergu M, Rădulescu VD: Bifurcation and asymptotics for the Lane-Emden-Fowler equation. Comptes Rendus Mathématique. Académie des Sciences. Paris 2003,337(4):259–264.

    Article  MathSciNet  MATH  Google Scholar 

  9. Lazer AC, McKenna PJ: On a singular nonlinear elliptic boundary-value problem. Proceedings of the American Mathematical Society 1991,111(3):721–730. 10.1090/S0002-9939-1991-1037213-9

    MathSciNet  Article  MATH  Google Scholar 

  10. Zhang Z: The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equation. Journal of Mathematical Analysis and Applications 2005,312(1):33–43. 10.1016/j.jmaa.2005.03.023

    MathSciNet  Article  MATH  Google Scholar 

  11. Zhang Z, Cheng J: Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems. Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2004,57(3):473–484.

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhang Z, Yu J: On a singular nonlinear Dirichlet problem with a convection term. SIAM Journal on Mathematical Analysis 2000,32(4):916–927. 10.1137/S0036141097332165

    MathSciNet  Article  MATH  Google Scholar 

  13. Aranda C, Godoy T: On a nonlinear Dirichlet problem with a singularity along the boundary. Differential and Integral Equations 2002,15(11):1313–1324.

    MathSciNet  MATH  Google Scholar 

  14. Cîrstea F-C, Rădulescu VD: Asymptotics for the blow-up boundary solution of the logistic equation with absorption. Comptes Rendus Mathématique. Académie des Sciences. Paris 2003,336(3):231–236.

    Article  MathSciNet  MATH  Google Scholar 

  15. Cîrstea F-C, Du Y: General uniqueness results and variation speed for blow-up solutions of elliptic equations. Proceedings of the London Mathematical Society. Third Series 2005,91(2):459–482. 10.1112/S0024611505015273

    MathSciNet  Article  MATH  Google Scholar 

  16. Resnick SI: Extreme Values, Regular Variation, and Point Processes, Applied Probability. A Series of the Applied Probability Trust. Volume 4. Springer, New York; 1987:xii+320.

    Google Scholar 

  17. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. 3rd edition. Springer, Berlin; 1998.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Informational Science, Yantai University, Yantai, Shandong, 264005, China

    Zhijun Zhang

  2. College of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China

    Jianning Yu

Authors
  1. Zhijun Zhang
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  2. Jianning Yu
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Correspondence to Zhijun Zhang.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Zhang, Z., Yu, J. The exact asymptotic behaviour of the unique solution to a singular Dirichlet problem. Bound Value Probl 2006, 75674 (2006). https://doi.org/10.1155/BVP/2006/75674

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  • DOI: https://doi.org/10.1155/BVP/2006/75674

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Unique Solution
  • Ordinary Differential Equation
  • Asymptotic Behaviour