Abstract and Applied Analysis

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

Yongju Yang and Xinguang Zhang

Full-text: Open access

Abstract

We study the existence of entire positive solutions for the semilinear elliptic system with quadratic gradient terms, Δ u i + | u i | 2 = p i ( | x | ) f i ( u 1 , u 2 , , u d ) for i = 1,2 , , d on R N , N 3 and d { 1,2 , 3 , } . We establish the conditions on p i that ensure the existence of nonnegative radial solutions blowing up at infinity and also the conditions for bounded solutions on the entire space. The condition on f i is simple and different to the Keller-Osserman condition.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 697565, 15 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365168872

Digital Object Identifier
doi:10.1155/2012/697565

Mathematical Reviews number (MathSciNet)
MR2999920

Zentralblatt MATH identifier
1257.35094

Citation

Yang, Yongju; Zhang, Xinguang. Entire Blow-Up Solutions of Semilinear Elliptic Systems with Quadratic Gradient Terms. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 697565, 15 pages. doi:10.1155/2012/697565. https://projecteuclid.org/euclid.aaa/1365168872


Export citation

References

  • L. Bieberbach, “$\Delta u={e}^{u}$ und die automorphen Funktionen,” Mathematische Annalen, vol. 77, no. 2, pp. 173–212, 1916.
  • J. B. Keller, “On solutions of $\Delta u=f(u)$,” Communications on Pure and Applied Mathematics, vol. 10, pp. 503–510, 1957.
  • R. Osserman, “On the inequality $\Delta u\geq f(u)$,” Pacific Journal of Mathematics, vol. 7, pp. 1641–1647, 1957.
  • 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.
  • 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.
  • K.-S. Cheng and W.-M. Ni, “On the structure of the conformal scalar curvature equation on ${R}^{n}$,” Indiana University Mathematics Journal, vol. 41, no. 1, pp. 261–278, 1992.
  • A. V. Lair and A. W. Wood, “Large solutions of semilinear elliptic problems,” Nonlinear Analysis A, vol. 37, no. 6, pp. 805–812, 1999.
  • A. V. Lair and A. W. Wood, “Large solutions of sublinear elliptic equations,” Nonlinear Analysis A, vol. 39, no. 6, pp. 745–753, 2000.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • D.-P. Covei, “Large and entire large solution for a quasilinear problem,” Nonlinear Analysis A, vol. 70, no. 4, pp. 1738–1745, 2009.
  • 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.
  • D.-P. Covei, “Schrödinger systems with a convection term for the $(p1,\ldots ,{p}_{d})$ -Laplacian in ${R}^{N}$,” Electronic Journal of Differential Equations, vol. 2012, no. 67, pp. 1–13, 2012.
  • R. Dalmasso, “Existence and uniqueness of positive solutions of semilinear elliptic systems,” Nonlinear Analysis A, vol. 39, no. 5, pp. 559–568, 2000.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • W. M. Ni, “On the elliptic equation $\Delta u= K(x){u}^{(n+2)/(n-2)}=0$, its generalizations, and applications in geometry,” Indiana University Mathematics Journal, vol. 31, no. 4, pp. 493–529, 1982.