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International Journal of Differential Equations
Volume 2010 (2010), Article ID 104625, 17 pages
http://dx.doi.org/10.1155/2010/104625
Research Article

Large Solutions of Quasilinear Elliptic System of Competitive Type: Existence and Asymptotic Behavior

1Institute of Mathematics, School of Mathematics Science, Nanjing Normal University, Jiangsu, Nanjing 210046, China
2College of Zhongbei, Nanjing Normal University, Jiangsu, Nanjing 210046, China

Received 22 July 2009; Accepted 23 October 2009

Academic Editor: Wenming Zou

Copyright © 2010 Lin Wei and Zuodong Yang. 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. G. Astrita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, NY, USA, 1974.
  2. L. K. Martinson and K. B. Pavlov, “Unsteady shear flows of a conducting fluid with a rheological power law,” Magnitnaya Gidrodinamika, vol. 2, pp. 50–58, 1971.
  3. A. S. Kalashnikov, “On a nonlinaer equation appearing in the theory of non-stationary filtration,” Trudy Seminara imeni I. G. Petrovskogo, vol. 4, pp. 137–146, 1978.
  4. Z. M. Guo, “Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 18, no. 10, pp. 957–971, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Z. M. Guo and J. R. L. Webb, “Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large,” Proceedings of the Royal Society of Edinburgh: Section A, vol. 124, no. 1, pp. 189–198, 1994. View at Zentralblatt MATH · View at MathSciNet
  6. Y. Du, “Effects of a degeneracy in the competition model. Part II. Perturbation and dynamical behaviour,” Journal of Differential Equations, vol. 181, no. 1, pp. 133–164, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. Y. Du, “Effects of a degeneracy in the competition model. Part I. Classical and generalized steady-state solutions,” Journal of Differential Equations, vol. 181, no. 1, pp. 92–132, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. E. N. Dancer and Y. Du, “Effects of certain degeneracies in the predator-prey model,” SIAM Journal on Mathematical Analysis, vol. 34, no. 2, pp. 292–314, 2002. View at Zentralblatt MATH · View at MathSciNet
  9. J. García-Melián, R. Letelier-Albornoz, and J. Sabina de Lis, “The solvability of an elliptic system under a singular boundary condition,” Proceedings of the Royal Society of Edinburgh: Section A, vol. 136, no. 3, pp. 509–546, 2006. View at Publisher · View at Google Scholar · View at Scopus
  10. J. López-Gómez, “Coexistence and meta-coexistence for competing species,” Houston Journal of Mathematics, vol. 29, no. 2, pp. 483–536, 2003. View at Zentralblatt MATH · View at MathSciNet
  11. J. García-Melián, A. Suárez, and J. Sabina de Lis, “Existence and uniqueness of positive large solutions to some cooperative elliptic systems,” Advanced Nonlinear Studies, vol. 3, pp. 193–206, 2003. View at Zentralblatt MATH
  12. S. Huang, Q. Tian, and C. Mu, “Large solutions of elliptic system of competitive type: existence uniqueness and asymptotic behavior,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, pp. 4544–4552, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. L. Bieberbach, “Δu=eu und die automorphen Funktionen,” Mathematische Annalen, vol. 77, no. 2, pp. 173–212, 1916. View at Publisher · View at Google Scholar · View at MathSciNet
  14. C. Bandle and M. Marcus, “Sur les solutions maximales de problèmes elliptiques nonlinèaires: bornes isopèrimètriques et comportement asymptotique,” Comptes Rendus de l'Acadèmie des Sciences. Sèrie I. Mathèmatique, vol. 311, no. 2, pp. 91–93, 1990. View at Zentralblatt MATH · View at MathSciNet
  15. C. Bandle and M. Marcus, “Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour,” Journal d'Analyse Mathématique, vol. 58, pp. 9–24, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. C. Bandle and M. Essèn, “On the solutions of quasilinear elliptic problems with boundary blow-up,” in Partial Differential Equations of Elliptic Type, vol. 35 of Symposia Mathematica, pp. 93–111, Cambridge University Press, Cambridge, UK, 1994. View at Zentralblatt MATH · View at MathSciNet
  17. C. Bandle and M. Marcus, “On second-order effects in the boundary behaviour of large solutions of semilinear elliptic problems,” Differential and Integral Equations, vol. 11, no. 1, pp. 23–34, 1998. View at Zentralblatt MATH · View at MathSciNet
  18. M. Chuaqui, C. Cortázar, M. Elgueta, C. Flores, R. Letelier, and J. García-Melián, “On an elliptic problem with boundary blow-up and a singular weight: the radial case,” Proceedings of the Royal Society of Edinburgh: Section A, vol. 133, no. 6, pp. 1283–1297, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. Chuaqui, C. Cortázar, and J. Garcia-Melián, “Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights,” Communications on Pure and Applied Analysis, vol. 3, no. 4, pp. 653–662, 2004. View at Publisher · View at Google Scholar · View at Scopus
  20. M. del Pino and R. Letelier, “The influence of domain geometry in boundary blow-up elliptic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 48, no. 6, pp. 897–904, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. G. Díaz and R. Letelier, “Explosive solutions of quasilinear elliptic equations: existence and uniqueness,” Nonlinear Analysis: Theory, Methods & Applications, vol. 20, no. 2, pp. 97–125, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Y. Du and Q. Huang, “Blow-up solutions for a class of semilinear elliptic and parabolic equations,” SIAM Journal on Mathematical Analysis, vol. 31, no. 1, pp. 1–18, 1999. View at Zentralblatt MATH · View at MathSciNet
  23. J. García-Melián, “A remark on the existence of large solutions via sub and supersolutions,” Electronic Journal of Differential Equations, vol. 2003, pp. 1–4, 2003. View at Zentralblatt MATH · View at Scopus
  24. J. García-Melián, R. Letelier-Albornoz, and J. Sabina de Lis, “Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up,” Proceedings of the American Mathematical Society, vol. 129, no. 12, pp. 3593–3602, 2001. View at Zentralblatt MATH · View at MathSciNet
  25. J. B. Keller, “On solution of δu=f(u),” Communications on Pure & Applied Mathematics, vol. 10, pp. 503–510, 1957. View at Publisher · View at Google Scholar
  26. C. Loewner and L. Nirenberg, “Partial differential equations invariant under conformal or projective transformations,” in Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), pp. 245–272, Academic Press, New York, NY, USA, 1974. View at Zentralblatt MATH · View at MathSciNet
  27. A. C. Lazer and P. J. McKenna, “On a problem of Bieberbach and Rademacher,” Nonlinear Analysis: Theory, Methods & Applications, vol. 21, no. 5, pp. 327–335, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. A. C. Lazer and P. J. McKenna, “Asymptotic behavior of solutions of boundary blowup problems,” Differential and Integral Equations, vol. 7, no. 3-4, pp. 1001–1019, 1994. View at MathSciNet
  29. V. A. Kondrat'ev and V. A. Nikishkin, “On the asymptotic behavior near the boundary of the solution of a singular boundary value problem for a semilinear elliptic equation,” Differential Equations, vol. 26, no. 3, pp. 465–468, 1990. View at MathSciNet
  30. M. Marcus and L. Véron, “Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 14, no. 2, pp. 237–274, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. A. Mohammed, G. Porcu, and G. Porru, “Large solutions to some non-linear O.D.E. with singular coefficients,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 1, pp. 513–524, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. R. Osserman, “On the inequality Δuf(u),” Pacific Journal of Mathematics, vol. 7, pp. 1641–1647, 1957. View at Zentralblatt MATH · View at MathSciNet
  33. L. Véron, “Semilinear elliptic equations with uniform blow-up on the boundary,” Journal d'Analyse Mathématique, vol. 59, pp. 231–250, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. Z. Zhang, “A remark on the existence of explosive solutions for a class of semilinear elliptic equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 41, no. 1-2, pp. 143–148, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. Z. D. Yang, B. Xu, and M. Wu, “Existence of positive boundary blow-up solutions for quasilinear elliptic equations via sub and supersolutions,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 492–498, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. Z. D. Yang, “Existence of explosive positive solutions of quasilinear elliptic equations,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 581–588, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  37. Z. D. Yang and Q. Lu, “Existence of entire explosive positive radial solutions of sublinear elliptic systems,” Communications in Nonlinear Science & Numerical Simulation, vol. 6, no. 2, pp. 88–92, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. Z. D. Yang, “Existence of entire explosive positive radial solutions of quasilinear elliptic systems,” International Journal of Mathematics and Mathematical Sciences, no. 46, pp. 2907–2927, 2003. View at Zentralblatt MATH · View at MathSciNet
  39. Z. D. Yang and H. Yang, “Existence of positive entire solutions for quasilinear elliptic systems,” Annals of Differential Equations, vol. 18, no. 4, pp. 417–424, 2002. View at MathSciNet
  40. J. García-Melián and J. D. Rossi, “Boundary blow-up solutions to elliptic systems of competitive type,” Journal of Differential Equations, vol. 206, no. 1, pp. 156–181, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. J. García-Melián, “A remark on uniqueness of large solutions for elliptic systems of competitive type,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 608–616, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. Z. D. Yang and M. Z. Wu, “Existence of boundary blow-up solutions for a class of quasilinear elliptic systems for the subcritical case,” Communications on Pure and Applied Analysis, vol. 6, no. 2, pp. 531–540, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. J. García-Melián, “Large solutions for an elliptic system of quasilinear equations,” Journal of Differential Equations, vol. 245, no. 12, pp. 3735–3752, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet