About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 697565, 15 pages
http://dx.doi.org/10.1155/2012/697565
Research Article

Entire Blow-Up Solutions of Semilinear Elliptic Systems with Quadratic Gradient Terms

1School of Mathematics and Statistics, Nanyang Normal University, Henan, Nanyang 473061, China
2School of Mathematical and Informational Sciences, Yantai University, Shandong, Yantai 264005, China

Received 1 September 2012; Accepted 8 November 2012

Academic Editor: Yong Hong Wu

Copyright © 2012 Yongju Yang and Xinguang Zhang. 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. 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
  2. J. B. Keller, “On solutions of Δu=f(u),” Communications on Pure and Applied Mathematics, vol. 10, pp. 503–510, 1957.
  3. R. Osserman, “On the inequality Δuf(u),” Pacific Journal of Mathematics, vol. 7, pp. 1641–1647, 1957.
  4. 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
  5. A. V. Lair, “A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 240, no. 1, pp. 205–218, 1999. View at Publisher · View at Google Scholar
  6. K.-S. Cheng and W.-M. Ni, “On the structure of the conformal scalar curvature equation on Rn,” Indiana University Mathematics Journal, vol. 41, no. 1, pp. 261–278, 1992. View at Publisher · View at Google Scholar
  7. A. V. Lair and A. W. Wood, “Large solutions of semilinear elliptic problems,” Nonlinear Analysis A, vol. 37, no. 6, pp. 805–812, 1999. View at Publisher · View at Google Scholar
  8. A. V. Lair and A. W. Wood, “Large solutions of sublinear elliptic equations,” Nonlinear Analysis A, vol. 39, no. 6, pp. 745–753, 2000. View at Publisher · View at Google Scholar
  9. A. V. Lair and A. W. Wood, “Existence of entire large positive solutions of semilinear elliptic systems,” Journal of Differential Equations, vol. 164, no. 2, pp. 380–394, 2000. View at Publisher · View at Google Scholar
  10. F.-C. Şt. Cîrstea and V. D. Rădulescu, “Entire solutions blowing up at infinity for semilinear elliptic systems,” Journal de Mathématiques Pures et Appliquées, vol. 81, no. 9, pp. 827–846, 2002. View at Publisher · View at Google Scholar
  11. M. Ghergu and V. Rădulescu, “Explosive solutions of semilinear elliptic systems with gradient term,” Revista de la Real Academia de Ciencias Exactas A, vol. 97, no. 3, pp. 437–445, 2003.
  12. Y. Peng and Y. Song, “Existence of entire large positive solutions of a semilinear elliptic system,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 687–698, 2004. View at Publisher · View at Google Scholar
  13. X. Zhang and L. Liu, “The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 300–308, 2010. View at Publisher · View at Google Scholar
  14. D.-P. Covei, “Radial and nonradial solutions for a semilinear elliptic system of Schrödinger type,” Funkcialaj Ekvacioj, vol. 54, no. 3, pp. 439–449, 2011. View at Publisher · View at Google Scholar
  15. D.-P. Covei, “Large and entire large solution for a quasilinear problem,” Nonlinear Analysis A, vol. 70, no. 4, pp. 1738–1745, 2009. View at Publisher · View at Google Scholar
  16. D.-P. Covei, “Existence of entire radically symmetric solutions for a quasilinear system with d-equations,” Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 3, pp. 433–439, 2011.
  17. D.-P. Covei, “Schrödinger systems with a convection term for the (p1,,pd) -Laplacian in RN,” Electronic Journal of Differential Equations, vol. 2012, no. 67, pp. 1–13, 2012.
  18. R. Dalmasso, “Existence and uniqueness of positive solutions of semilinear elliptic systems,” Nonlinear Analysis A, vol. 39, no. 5, pp. 559–568, 2000. View at Publisher · View at Google Scholar
  19. M. Ghergu and V. D. Rădulescu, Singular Elliptic Equations: Bifurcation and Asymptotic Analysis, vol. 37 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, UK, 2008.
  20. S. Chen and G. Lu, “Existence and nonexistence of positive radial solutions for a class of semilinear elliptic system,” Nonlinear Analysis A, vol. 38, no. 7, pp. 919–932, 1999. View at Publisher · View at Google Scholar
  21. M. Ghergu, C. Niculescu, and V. Rădulescu, “Explosive solutions of elliptic equations with absorption and non-linear gradient term,” Indian Academy of Sciences, vol. 112, no. 3, pp. 441–451, 2002. View at Publisher · View at Google Scholar
  22. J. Serrin and H. Zou, “Existence of positive entire solutions of elliptic Hamiltonian systems,” Communications in Partial Differential Equations, vol. 23, no. 3-4, pp. 577–599, 1998. View at Publisher · View at Google Scholar
  23. A. V. Lair and A. W. Wood, “Entire solution of a singular semilinear elliptic problem,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 498–505, 1996. View at Publisher · View at Google Scholar
  24. P. Quittner, “Blow-up for semilinear parabolic equations with a gradient term,” Mathematical Methods in the Applied Sciences, vol. 14, no. 6, pp. 413–417, 1991. View at Publisher · View at Google Scholar
  25. C. Bandle and E. Giarrusso, “Boundary blow up for semilinear elliptic equations with nonlinear gradient terms,” Advances in Differential Equations, vol. 1, no. 1, pp. 133–150, 1996.
  26. C. S. Yarur, “Existence of continuous and singular ground states for semilinear elliptic systems,” Electronic Journal of Differential Equations, vol. 1, pp. 1–27, 1998.
  27. X. Wang and A. W. Wood, “Existence and nonexistence of entire positive solutions of semilinear elliptic systems,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 361–368, 2002. View at Publisher · View at Google Scholar
  28. X. Li, J. Zhang, S. Lai, and Y. Wu, “The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system,” Journal of Differential Equations, vol. 250, no. 4, pp. 2197–2226, 2011. View at Publisher · View at Google Scholar
  29. S. Lai and Y. Wu, “Global solutions and blow-up phenomena to a shallow water equation,” Journal of Differential Equations, vol. 249, no. 3, pp. 693–706, 2010. View at Publisher · View at Google Scholar
  30. X. Li, Y. Wu, and S. Lai, “A sharp threshold of blow-up for coupled nonlinear Schrödinger equations,” Journal of Physics A, vol. 43, no. 16, article 165205, p. 11, 2010. View at Publisher · View at Google Scholar
  31. J. Zhang, X. Li, and Y. H. Wu, “Remarks on the blow-up rate for critical nonlinear Schrödinger equation with harmonic potential,” Applied Mathematics and Computation, vol. 208, pp. 389–396, 2009.
  32. X. Zhang, “A necessary and sufficient condition for the existence of large solutions to “mixed” type elliptic systems,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2359–2364, 2012. View at Publisher · View at Google Scholar
  33. W. M. Ni, “On the elliptic equation Δu=K(x)u(n+2)/(n-2) = 0, its generalizations, and applications in geometry,” Indiana University Mathematics Journal, vol. 31, no. 4, pp. 493–529, 1982. View at Publisher · View at Google Scholar