Abstract
In the present paper we establish the existence of three positive weak solutions for a quasilinear elliptic problem involving a singular term of the type \({u^{-\gamma}}\). As far as we know this is the first contribution in the higher dimensional case for arbitrary \({\gamma > 0}\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Canino A., Degiovanni M.: A variational approach to a class of singular semilinear elliptic equations. J. Convex Anal. 11, 147–162 (2004)
Cupini G., Marcellini P., Mascolo E.: Local boundedness of solutions to quasilinear elliptic systems. Manuscripta Math. 137, 287–315 (2012)
di Benedetto E.: \({C^{1+\alpha}}\)- Local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1983)
Faraci, F., Smyrlis, G.: Three solutions for a singular quasilinear elliptic problem, (submitted)
Giacomoni J., Schindler I., Takàč P.: Sobolev versus Hölder minimizers and global multiplicity for a singular and quasilinear equation. Annali Scuola Normale Superiore Pisa, Cl. Sci. 6, 117–158 (2007)
Hirano N., Saccon C., Shioji N.: Brezis—Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem. J. Differential Equations. 245, 1997–2037 (2008)
Perera K., Silva E.A.B.: Existence and multiplicity of positive solutions for singular quasilinear problems. J. Math. Anal. Appl. 323, 1238–1252 (2006)
Perera K., Zhang Z.: Multiple positive solutions of singular p -Laplacian problems by variational methods. Bound. Value Probl. 3, 377–382 (2005)
Ricceri B.: Sublevel sets and global minima of coercive functionals and local minima of their perturbation. J. Nonlinear Convex Anal. 5, 157–168 (2004)
Ricceri B.: A further three critical points theorem. Nonlinear Anal. 71, 4151–4157 (2009)
Szulkin A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problem. Ann. Inst. H. Poincaré. 3, 77–109 (1986)
Zhao L., He Y., Zhao P.: The existence of three positive solutions of a singular p-Laplacian problem. Nonlinear Anal. 74, 5745–5753 (2011)
Zhang Z.: points and positive solutions of singular elliptic boundary value problems. J. Math. Anal. Appl. 302, 476–483 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Faraci, F., Smyrlis, G. Three solutions for a class of higher dimensional singular problems. Nonlinear Differ. Equ. Appl. 23, 45 (2016). https://doi.org/10.1007/s00030-016-0398-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-016-0398-6