International Journal of Differential Equations

Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

Faten Toumi

Full-text: Open access

Abstract

This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problem Δu=λp(x)f(u) in Ω subject to some Dirichlet conditions, where Ω is a regular domain in n(n3) with compact boundary. The nonlinearity f is nonnegative continuous and the potential p belongs to some Kato class K(Ω). So we prove the existence of a positive continuous solution depending on λ by the use of a potential theory approach.

Article information

Source
Int. J. Differ. Equ., Volume 2010 (2010), Article ID 134078, 10 pages.

Dates
Received: 18 June 2010
Accepted: 25 September 2010
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485399926

Digital Object Identifier
doi:10.1155/2010/134078

Mathematical Reviews number (MathSciNet)
MR2746095

Zentralblatt MATH identifier
1221.35160

Citation

Toumi, Faten. Existence of Positive Bounded Solutions of Semilinear Elliptic Problems. Int. J. Differ. Equ. 2010 (2010), Article ID 134078, 10 pages. doi:10.1155/2010/134078. https://projecteuclid.org/euclid.ijde/1485399926


Export citation

References

  • A. L. Edelson, “Entire solutions of singular elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 139, no. 2, pp. 523–532, 1989.
  • A. V. Lair and A. W. Shaker, “Entire solution of a singular semilinear elliptic problem,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 498–505, 1996.
  • A. C. Lazer and P. J. McKenna, “On a singular nonlinear elliptic boundary-value problem,” Proceedings of the American Mathematical Society, vol. 111, no. 3, pp. 721–730, 1991.
  • H. Mâagli and M. Zribi, “Existence and estimates of solutions for singular nonlinear elliptic problems,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 522–542, 2001.
  • N. Zeddini, “Positive solutions for a singular nonlinear problem on a bounded domain in ${\mathbb{R}}^{2}$,” Potential Analysis, vol. 18, no. 2, pp. 97–118, 2003.
  • J. Glover and P. J. McKenna, “Solving semilinear partial differential equations with probabilistic potential theory,” Transactions of the American Mathematical Society, vol. 290, no. 2, pp. 665–681, 1985.
  • Z. M. Ma and R. M. Song, “Probabilistic methods in Schrödinger equations,” in Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989), E. Cinlar, K. L. Chung, and R. K. Getoor, Eds., vol. 18 of Progress in Probability, pp. 135–164, Birkhäuser Boston, Boston, Mass, USA, 1990.
  • Z. Q. Chen, R. J. Williams, and Z. Zhao, “On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions,” Mathematische Annalen, vol. 298, no. 3, pp. 543–556, 1994.
  • S. Athreya, “On a singular semilinear elliptic boundary value problem and the boundary Harnack principle,” Potential Analysis, vol. 17, no. 3, pp. 293–301, 2002.
  • K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger's Equation, vol. 312 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1995.
  • Z. X. Zhao, “On the existence of positive solutions of nonlinear elliptic equations–-a probabilistic potential theory approach,” Duke Mathematical Journal, vol. 69, no. 2, pp. 247–258, 1993.
  • H. Maâgli, “Perturbation semi-linéaire des résolvantes et des semi-groupes,” Potential Analysis, vol. 3, pp. 61–87, 1994.
  • D. V. Widder, The Laplace Transform, vol. 6 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1941.
  • I. Bachar, H. Maâgli, and N. Zeddini, “Estimates on the Green function and existence of positive solutions of nonlinear singular elliptic equations,” Communications in Contemporary Mathematics, vol. 5, no. 3, pp. 401–434, 2003.
  • H. Mâagli and M. Zribi, “On a new Kato class and singular solutions of a nonlinear elliptic equation in bounded domains of ${\mathbb{R}}^{n}$,” Positivity, vol. 9, no. 4, pp. 667–686, 2005.
  • F. Toumi, “Existence of positive solutions for nonlinear boundary value problems in bounded domains of ${\mathbb{R}}^{n}$,” Abstract and Applied Analysis, vol. 2006, Article ID 95480, 18 pages, 2006.
  • F. Toumi and N. Zeddini, “Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of ${\mathbb{R}}^{n}$,” Electronic Journal of Differential Equations, vol. 2005, 14 pages, 2005.
  • S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory. Probability and Mathematical Statistic, Academic Press, New York, NY, USA, 1978.
  • D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer Monographs in Mathematics, Springer, London, UK, 2001.
  • K. Hirata, “On the existence of positive solutions of singular nonlinear elliptic equations with Dirichlet boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 885–891, 2008.