Skip to content


  • Research Article
  • Open Access

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

  • 1Email author and
  • 2
Boundary Value Problems20062006:79679

  • Received: 18 November 2005
  • Accepted: 18 April 2006
  • Published:


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.


  • Differential Equation
  • Continuous Function
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation


Authors’ Affiliations

Departamento de Matemática-CCEN, Universidade Federal do Pará, Belém-Pará, 66.075-110, Brazil
Dedicated to our dear friend and collaborator Professor Claudianor O. Alves, Brazil


  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.MathSciNetMATHGoogle 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.2005.01.008MathSciNetView ArticleMATHGoogle Scholar
  3. Chabrowski J, Yang J: Existence theorems for elliptic equations involving supercritical Sobolev exponent. Advances in Differential Equations 1997,2(2):231–256.MathSciNetMATHGoogle Scholar
  4. Chipot M: Elements of Nonlinear Analysis, Birkhäuser Advanced Texts: Basel Textbooks. Birkhäuser, Basel; 2000:viii+256.View ArticleGoogle 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)00169-7MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle Scholar
  7. Corrêa FJSA: On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Analysis. Theory, Methods & Applications 2004,59(7):1147–1155.MathSciNetView ArticleMATHGoogle 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.Google Scholar
  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_13Google 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)00740-3MathSciNetView ArticleMATHGoogle 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.2001.7696MathSciNetView ArticleMATHGoogle 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)80118-0MathSciNetView ArticleMATHGoogle 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.MATHGoogle 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.View ArticleMATHGoogle 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.3160130308MathSciNetView ArticleMATHGoogle 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.2005.03.006MathSciNetView ArticleMATHGoogle Scholar
  19. Rabinowitz PH: Variational methods for nonlinear elliptic eigenvalue problems. Indiana University Mathematics Journal 1974, 23: 729–754. 10.1512/iumj.1974.23.23061MathSciNetView ArticleMATHGoogle 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.1998.3535MathSciNetView ArticleMATHGoogle Scholar
  21. Stańczy R: Nonlocal elliptic equations. Nonlinear Analysis. Theory, Methods & Applications 2001,47(5):3579–3584. 10.1016/S0362-546X(01)00478-3MathSciNetView ArticleMATHGoogle Scholar


© F. J. S. A. Corrêa and G. M. Figueiredo 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.