Skip to main content
Boundary Value Problems
Download PDF

Radial solutions for a nonlocal boundary value problem

Boundary Value Problems volume 2006, Article number: 32950 (2006) Cite this article

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]

References

  1. 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.

    MathSciNet  MATH  Google Scholar 

  2. 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.

    MathSciNet  Google Scholar 

  3. 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. 4.

    Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985:xiv+450.

    Book  MATH  Google Scholar 

  5. 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.

    MathSciNet  MATH  Google Scholar 

  6. 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/S0308210500021727

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Gomes JM, Sanchez L: On a variational approach to some non-local boundary value problems. Applicable Analysis 2005,84(9):909-925. 10.1080/00036810500048202

    MathSciNet  Article  MATH  Google Scholar 

  8. 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.

    MathSciNet  MATH  Google Scholar 

  9. 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)00201-6

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York; 1992:xvi+777.

    MATH  Google Scholar 

  11. 11.

    Smirnov V: Cours de Mathématiques Supérieures. Volume 2. Mir, Moscoú; 1970.

    Google Scholar 

  12. 12.

    Walter W: Ordinary Differential Equations, Graduate Texts in Mathematics. Volume 182. Springer, New York; 1998:xii+380.

    Google Scholar 

  13. 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.2005.04.030

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Zeidler E: Nonlinear Functional Analysis and Its Applications—I : Fixed-Point Theorems. Springer, New York; 1986:xxi+897.

    Book  MATH  Google Scholar 

Download references

Author information

Affiliations

  1. Área Científica de Matemática, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro, Lisboa, 1-1950-062, Portugal

    Ricardo Enguiça

  2. Faculdade de Ciências da Universidade de Lisboa, Avenida Professor Gama Pinto 2, Lisboa, 1649-003, Portugal

    Luís Sanchez

Authors
  1. Ricardo Enguiça
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luís Sanchez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luís Sanchez.

Rights and permissions

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.

Reprints and Permissions

About this article

Cite this article

Enguiça, R., Sanchez, L. Radial solutions for a nonlocal boundary value problem. Bound Value Probl 2006, 32950 (2006). https://doi.org/10.1155/BVP/2006/32950

Download citation

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Point Theorem