Boundary Value Problems

Impact Factor 0.819

Boundary Value Problems Cover Image
Research Article
Open Access

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

Boundary Value Problems20062006:75674

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

©  Z. Zhang and J. Yu 2006

Received: 23 September 2005

Accepted: 15 November 2005

Published: 23 May 2006

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.

[1234567891011121314151617]

Authors’ Affiliations

(1)
Department of Mathematics and Informational Science, Yantai University
(2)
College of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University

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. 10.1080/03605307708820029MathSciNetView ArticleMATHGoogle Scholar
  2. Fulks W, Maybee JS: A singular non-linear equation. Osaka Journal of Mathematics 1960, 12: 1–19.MathSciNetMATHGoogle 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. 10.1137/0138024MathSciNetView ArticleMATHGoogle Scholar
  4. Stuart CA: Existence and approximation of solutions of non-linear elliptic equations. Mathematische Zeitschrift 1976,147(1):53–63. 10.1007/BF01214274MathSciNetView ArticleMATHGoogle 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.View ArticleMathSciNetMATHGoogle 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/BF01442856MathSciNetView ArticleMATHGoogle 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.View ArticleMathSciNetMATHGoogle 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.View ArticleMathSciNetMATHGoogle 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/S0002-9939-1991-1037213-9MathSciNetView ArticleMATHGoogle 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.2005.03.023MathSciNetView ArticleMATHGoogle 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.View ArticleMathSciNetMATHGoogle 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/S0036141097332165MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle 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.View ArticleMathSciNetMATHGoogle 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/S0024611505015273MathSciNetView ArticleMATHGoogle 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.MATHGoogle Scholar

Copyright

© Z. Zhang and J. Yu 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.