Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012, Article ID 137379, 16 pages
http://dx.doi.org/10.1155/2012/137379
Research Article

Existence of Solutions of a Nonlocal Elliptic System via Galerkin Method

1Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2Unidade Acadêmica de Matemática e Estatística, Centro de Ciências e Tecnologia, Universidade Federal de Campina Grande, 58.109-979 Campina Grande, Brazil

Received 2 February 2012; Accepted 28 February 2012

Academic Editor: Pavel Drábek

Copyright © 2012 Alberto Cabada and Francisco Julio S. A. Corrêa. 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.

Linked References

  1. A. Cabada, J. A. Cid, and L. Sanchez, “Existence of solutions for elliptic systems with nonlocal terms in one dimension,” Boundary Value Problems, vol. 2011, Article ID 518431, 12 pages, 2011. View at Publisher · View at Google Scholar
  2. R. P. Agarwal, M. Bohner, and V. B. Shakhmurov, “Linear and nonlinear nonlocal boundary value problems for differential-operator equations,” Applicable Analysis, vol. 85, no. 6-7, pp. 701–716, 2006. View at Publisher · View at Google Scholar
  3. W. Allegretto and A. Barabanova, “Existence of positive solutions of semilinear elliptic equations with nonlocal terms,” Funkcialaj Ekvacioj, vol. 40, no. 3, pp. 395–409, 1997. View at Google Scholar
  4. W. Allegretto and A. Barabanova, “Positivity of solutions of elliptic equations with nonlocal terms,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 126, no. 3, pp. 643–663, 1996. View at Publisher · View at Google Scholar
  5. P. Freitas, “A nonlocal Sturm-Liouville eigenvalue problem,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 124, no. 1, pp. 169–188, 1994. View at Publisher · View at Google Scholar
  6. D. Liang, J. W.-H. So, F. Zhang, and X. Zou, “Population dynamic models with nonlocal delay on bounded domains and their numerical computations,” Differential Equations and Dynamical Systems, vol. 11, no. 1-2, pp. 117–139, 2003. View at Google Scholar
  7. S. Pečiulyte, O. Štikoniene, and A. Štikonas, “Sturm-Liouville problem for stationary differential operator with nonlocal integral boundary condition,” Mathematical Modelling and Analysis, vol. 10, no. 4, pp. 377–392, 2005. View at Google Scholar
  8. N. Sergejeva, “Fučik spectrum for the second order BVP with nonlocal boundary condition,” Nonlinear Analysis. Modelling and Control, vol. 12, no. 3, pp. 419–429, 2007. View at Google Scholar
  9. H. L. Tidke and M. B. Dhakne, “Existence and uniqueness of mild solutions of second order Volterra integrodifferential equations with nonlocal conditions,” Applied Mathematics E-Notes, vol. 9, pp. 101–108, 2009. View at Google Scholar
  10. F. J. S. A. Corrêa, M. Delgado, and A. Suárez, “Some non-local population models with non-linear diffusion,” Revista Integración. Temas de Matemáticas, Universidade Industrial de Santander, vol. 28, no. 1, pp. 37–49, 2010. View at Google Scholar
  11. N. Dodds, Non-local differential equations, Doctoral thesis, Division of Mathematics, University of Dundee, Dundee, Scotland, 2005.
  12. P. Freitas, “Nonlocal reaction-diffusion equations,” in Differential Equations with Applications to Biology, vol. 21 of Fields Institute Communications, pp. 187–204, American Mathematical Society, Providence, RI, USA, 1999. View at Google Scholar
  13. J. López-Gómez, “On the structure and stability of the set of solutions of a nonlocal problem modeling Ohmic heating,” Journal of Dynamics and Differential Equations, vol. 10, no. 4, pp. 537–566, 1998. View at Publisher · View at Google Scholar
  14. A. Cabada and S. Lois, “Existence results for nonlinear problems with separated boundary conditions,” Nonlinear Analysis, vol. 35, no. 4, pp. 449–456, 1999. View at Publisher · View at Google Scholar
  15. D. G. de Figueiredo and E. Mitidieri, “A maximum principle for an elliptic system and applications to semilinear problems,” SIAM Journal on Mathematical Analysis, vol. 17, no. 4, pp. 836–849, 1986. View at Publisher · View at Google Scholar
  16. F. J. S. D. A. Corrêa, “On the existence of steady-state solutions of a reaction-diffusion system,” Nonlinear Times and Digest, vol. 2, no. 1, pp. 95–106, 1995. View at Google Scholar
  17. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Gauthier-Villars, Paris, France, 1969.
  18. H. Brezis and S. Kamin, “Sublinear elliptic equations in ,” Manuscripta Mathematica, vol. 74, no. 1, pp. 87–106, 1992. View at Publisher · View at Google Scholar
  19. H. Brezis and L. Oswald, “Remarks on sublinear elliptic equations,” Nonlinear Analysis, vol. 10, no. 1, pp. 55–64, 1986. View at Publisher · View at Google Scholar
  20. A. Ambrosetti, H. Brezis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,” Journal of Functional Analysis, vol. 122, no. 2, pp. 519–543, 1994. View at Publisher · View at Google Scholar
  21. X.-L. Fan and Q.-H. Zhang, “Existence of solutions for p(x)-Laplacian Dirichlet problem,” Nonlinear Analysis, vol. 52, no. 8, pp. 1843–1852, 2003. View at Publisher · View at Google Scholar
  22. X. Fan and D. Zhao, “On the spaces Lp(x) and Wm,p(x),” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 424–446, 2001. View at Publisher · View at Google Scholar