Skip to main content
Boundary Value Problems
Download PDF
  • Research Article
  • Open access
  • Published:

On the existence of positive solution for an elliptic equation of Kirchhoff type via Moser iteration method

Boundary Value Problems volume 2006, Article number: 79679 (2006) Cite this article

Abstract

We investigate the questions of existence of positive solution for the nonlocal problem and on, where is a bounded smooth domain of, and and are continuous functions.

[123456789101112131415161718192021]

References

  1. Alves CO, Corrêa FJSA: On existence of solutions for a class of problem involving a nonlinear operator. Communications on Applied Nonlinear Analysis 2001,8(2):43–56.

    MathSciNet  MATH  Google Scholar 

  2. Alves CO, Corrêa FJSA, Ma TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Computers & Mathematics with Applications 2005,49(1):85–93. 10.1016/j.camwa.2005.01.008

    Article  MathSciNet  MATH  Google Scholar 

  3. Chabrowski J, Yang J: Existence theorems for elliptic equations involving supercritical Sobolev exponent. Advances in Differential Equations 1997,2(2):231–256.

    MathSciNet  MATH  Google Scholar 

  4. Chipot M: Elements of Nonlinear Analysis, Birkhäuser Advanced Texts: Basel Textbooks. Birkhäuser, Basel; 2000:viii+256.

    Book  Google Scholar 

  5. Chipot M, Lovat B: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Analysis. Theory, Methods & Applications 1997,30(7):4619–4627. 10.1016/S0362-546X(97)00169-7

    Article  MathSciNet  MATH  Google Scholar 

  6. Chipot M, Rodrigues J-F: On a class of nonlocal nonlinear elliptic problems. RAIRO Modélisation Mathématique et Analyse Numérique 1992,26(3):447–467.

    MathSciNet  MATH  Google Scholar 

  7. Corrêa FJSA: On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Analysis. Theory, Methods & Applications 2004,59(7):1147–1155.

    Article  MathSciNet  MATH  Google Scholar 

  8. Corrêa FJSA, Menezes SDB: Existence of solutions to nonlocal and singular elliptic problems via Galerkin method. Electronic Journal of Differential Equations 2004, (19):1–10.

  9. Corrêa FJSA, Menezes SDB: Positive solutions for a class of nonlocal elliptic problems. In Contributions to Nonlinear Analysis, Progress in Nonlinear Differential Equations and Their Applications. Volume 66. Birkhäuser, Basel; 2006:195–206. 10.1007/3-7643-7401-2_13

    Google Scholar 

  10. Corrêa FJSA, Menezes SDB, Ferreira J: On a class of problems involving a nonlocal operator. Applied Mathematics and Computation 2004,147(2):475–489. 10.1016/S0096-3003(02)00740-3

    Article  MathSciNet  MATH  Google Scholar 

  11. Deng W, Duan Z, Xie C: The blow-up rate for a degenerate parabolic equation with a non-local source. Journal of Mathematical Analysis and Applications 2001,264(2):577–597. 10.1006/jmaa.2001.7696

    Article  MathSciNet  MATH  Google Scholar 

  12. Deng W, Li Y, Xie C: Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations. Applied Mathematics Letters 2003,16(5):803–808. 10.1016/S0893-9659(03)80118-0

    Article  MathSciNet  MATH  Google Scholar 

  13. Figueiredo GM: Multiplicidade de soluções positivas para uma classe de problemas quasilineares, Doct. dissertation. UNICAMP, São Paulo; 2004.

    Google Scholar 

  14. Kirchhoff G: Mechanik. Teubner, Leipzig; 1883.

    MATH  Google Scholar 

  15. Lions J-L: On some questions in boundary value problems of mathematical physics. In Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Rio de Janeiro, 1977), North-Holland Math. Stud.. Volume 30. North-Holland, Amsterdam; 1978:284–346.

    Google Scholar 

  16. Ma TF: Remarks on an elliptic equation of Kirchhoff type. Nonlinear Analysis. Theory, Methods & Applications 2005,63(5–7):e1967-e1977.

    Article  MATH  Google Scholar 

  17. Moser J: A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations. Communications on Pure and Applied Mathematics 1960, 13: 457–468. 10.1002/cpa.3160130308

    Article  MathSciNet  MATH  Google Scholar 

  18. Perera K, Zhang Z: Nontrivial solutions of Kirchhoff-type problems via the Yang index. Journal of Differential Equations 2006,221(1):246–255. 10.1016/j.jde.2005.03.006

    Article  MathSciNet  MATH  Google Scholar 

  19. Rabinowitz PH: Variational methods for nonlinear elliptic eigenvalue problems. Indiana University Mathematics Journal 1974, 23: 729–754. 10.1512/iumj.1974.23.23061

    Article  MathSciNet  MATH  Google Scholar 

  20. Souplet P: Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. Journal of Differential Equations 1999,153(2):374–406. 10.1006/jdeq.1998.3535

    Article  MathSciNet  MATH  Google Scholar 

  21. Stańczy R: Nonlocal elliptic equations. Nonlinear Analysis. Theory, Methods & Applications 2001,47(5):3579–3584. 10.1016/S0362-546X(01)00478-3

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática-CCEN, Universidade Federal do Pará, Belém-Pará, 66.075-110, Brazil

    Francisco Júlio S A Corrêa

  2. Dedicated to our dear friend and collaborator Professor Claudianor O. Alves, Brazil

    Giovany M Figueiredo

Authors
  1. Francisco Júlio S A Corrêa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giovany M Figueiredo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Júlio S A Corrêa.

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

Corrêa, F.J.S.A., Figueiredo, G.M. On the existence of positive solution for an elliptic equation of Kirchhoff type via Moser iteration method. Bound Value Probl 2006, 79679 (2006). https://doi.org/10.1155/BVP/2006/79679

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/BVP/2006/79679

Keywords