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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.

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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.

    MathSciNet  MATH  Google 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.

    MathSciNet  Google 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.

    Book  MATH  Google 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.

    MathSciNet  MATH  Google 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/S0308210500021727

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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)00201-6

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google 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.2005.04.030

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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Authors and 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
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  2. Luís Sanchez
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Correspondence to Luís Sanchez.

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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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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

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  • DOI: https://doi.org/10.1155/BVP/2006/32950

Keywords

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