Open Access

Radial solutions for a nonlocal boundary value problem

Boundary Value Problems20062006:32950

DOI: 10.1155/BVP/2006/32950

Received: 23 August 2005

Accepted: 22 December 2005

Published: 8 June 2006

Abstract

We consider the boundary value problem for the nonlinear Poisson equation with a nonlocal term , . We prove the existence of a positive radial solution when grows linearly in , using Krasnoselskiiés fixed point theorem together with eigenvalue theory. In presence of upper and lower solutions, we consider monotone approximation to solutions.

[1234567891011121314]

Authors’ Affiliations

(1)
Área Científica de Matemática, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro
(2)
Faculdade de Ciências da Universidade de Lisboa

References

  1. Bebernes JW, Lacey AA: Global existence and finite-time blow-up for a class of nonlocal parabolic problems. Advances in Differential Equations 1997,2(6):927-953.MathSciNetMATHGoogle Scholar
  2. Chang N-H, Chipot M: On some mixed boundary value problems with nonlocal diffusion. Advances in Mathematical Sciences and Applications 2004,14(1):1-24.MathSciNetGoogle Scholar
  3. De Coster C, Habets P: The lower and upper solutions method for boundary value problems. In Handbook of Differential Equations. Elsevier, New York; North-Holland, Amsterdam; 2004:69-160.Google Scholar
  4. Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985:xiv+450.View ArticleMATHGoogle Scholar
  5. Fijałkowski P, Przeradzki B: On a radial positive solution to a nonlocal elliptic equation. Topological Methods in Nonlinear Analysis 2003,21(2):293-300.MathSciNetMATHGoogle Scholar
  6. Freitas P, Sweers G: Positivity results for a nonlocal elliptic equation. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1998,128(4):697-715. 10.1017/S0308210500021727MathSciNetView ArticleMATHGoogle Scholar
  7. Gomes JM, Sanchez L: On a variational approach to some non-local boundary value problems. Applicable Analysis 2005,84(9):909-925. 10.1080/00036810500048202MathSciNetView ArticleMATHGoogle Scholar
  8. Gaudenzi M, Habets P, Zanolin F: Positive solutions of singular boundary value problems with indefinite weight. Bulletin of the Belgian Mathematical Society. Simon Stevin 2002,9(4):607-619.MathSciNetMATHGoogle Scholar
  9. Jiang D, Gao W, Wan A: A monotone method for constructing extremal solutions to fourth-order periodic boundary value problems. Applied Mathematics and Computation 2002,132(2-3):411-421. 10.1016/S0096-3003(01)00201-6MathSciNetView ArticleMATHGoogle Scholar
  10. Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York; 1992:xvi+777.MATHGoogle Scholar
  11. Smirnov V: Cours de Mathématiques Supérieures. Volume 2. Mir, Moscoú; 1970.Google Scholar
  12. Walter W: Ordinary Differential Equations, Graduate Texts in Mathematics. Volume 182. Springer, New York; 1998:xii+380.Google Scholar
  13. Yang Z: Positive solutions to a system of second-order nonlocal boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 2005,62(7):1251-1265. 10.1016/j.na.2005.04.030MathSciNetView ArticleMATHGoogle Scholar
  14. Zeidler E: Nonlinear Functional Analysis and Its Applications—I : Fixed-Point Theorems. Springer, New York; 1986:xxi+897.View ArticleMATHGoogle Scholar

Copyright

© Enguiça and Sanchez 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.