Abstract and Applied Analysis

Existence and Nonexistence of Positive Solutions for Quasilinear Elliptic Problem

K. Saoudi

Full-text: Open access

Abstract

Using variational arguments we prove some existence and nonexistence results for positive solutions of a class of elliptic boundary-value problems involving the p -Laplacian.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 275748, 9 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365174063

Digital Object Identifier
doi:10.1155/2012/275748

Mathematical Reviews number (MathSciNet)
MR2955036

Zentralblatt MATH identifier
1250.35086

Citation

Saoudi, K. Existence and Nonexistence of Positive Solutions for Quasilinear Elliptic Problem. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 275748, 9 pages. doi:10.1155/2012/275748. https://projecteuclid.org/euclid.aaa/1365174063


Export citation

References

  • V. Rădulescu and D. Repovš, “Combined effects in nonlinear problems arising in the study of anisotropic continuous media,” Nonlinear Analysis, vol. 75, no. 3, pp. 1524–1530, 2012.
  • C. A. Santos, “Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms,” Nonlinear Analysis, vol. 72, no. 9-10, pp. 3813–3819, 2010.
  • M. Cuesta and P. Takáč, “A strong comparison principle for positive solutions of degenerate elliptic equations,” Differential and Integral Equations, vol. 13, no. 4–6, pp. 721–746, 2000.
  • P. Lindqvist, “On the equation ${\text{div}(\vert \nabla u\vert }^{p-2}{\nabla u)+\lambda \vert u\vert }^{p-2}u=0$,” Proceedings of the American Mathematical Society, vol. 109, no. 1, pp. 157–164, 1990.
  • A. Anane, “Simplicité et isolation de la première valeur propre du $p$-laplacien avec poids,” Comptes Rendus des Séances de l'Académie des Sciences I, vol. 305, no. 16, pp. 725–728, 1987.
  • R. Filippucci, P. Pucci, and V. Rădulescu, “Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions,” Communications in Partial Differential Equations, vol. 33, no. 4–6, pp. 706–717, 2008.
  • P. Pucci and R. Servadei, “Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations,” Indiana University Mathematics Journal, vol. 57, no. 7, pp. 3329–3363, 2008.
  • E. DiBenedetto, “${C}^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations,” Nonlinear Analysis, vol. 7, no. 8, pp. 827–850, 1983.
  • P. Pucci and J. Serrin, “Maximum principles for elliptic partial differential equations,” in Handbook of Differential Equations: Stationary Partial Differential Equations, M. Chipot, Ed., vol. 4, pp. 355–483, Elsevier, 2007.