Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-09T16:42:36.608Z Has data issue: false hasContentIssue false

Three Solutions for a Singular Quasilinear Elliptic Problem

Published online by Cambridge University Press:  14 September 2018

Francesca Faraci
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy ([email protected])
George Smyrlis
Affiliation:
Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece ([email protected])

Abstract

In the present paper we deal with a quasilinear problem involving a singular term. By combining truncation techniques with variational methods, we prove the existence of three weak solutions. As far as we know, this is the first contribution in this direction in the high-dimensional case.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ambrosetti, A. and Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349381.Google Scholar
2.Arcoya, D. and Moreno-Mérida, L., Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity, Nonlinear Anal. 95 (2014), 281291.Google Scholar
3.Coclite, M. and Palmieri, G., On a singular nonlinear Dirichlet problem, Comm. Partial Differ. Equ. 14 (1989), 13151327.Google Scholar
4.Giacomoni, J. and Saoudi, K., $W^{1,p}_{0}$ versus C 1 local minimizers for a singular and critical functional, J. Math. Anal. Appl. 363 (2010), 697710.Google Scholar
5.Giacomoni, J., Schindler, I. and Takàč, P., Sobolev versus Hölder minimizers and global multiplicity for a singular and quasilinear equation, Annali Scuola Normale Superiore Pisa Cl. Sci. 6 (1) (2007), 117158.Google Scholar
6.Hirano, N., Saccon, C. and Shioji, N., Brezis–Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differ. Equ. 245 (2008), 19972037.Google Scholar
7.Lair, A. and Shaker, A., Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl. 211 (1997), 371385.Google Scholar
8.Lazer, A. and McKenna, P. J., On a singular nonlinear elliptic boundary value problem, Proc. AMS 111 (1991), 721730.Google Scholar
9.Perera, K. and Silva, E. A. B., Existence and multiplicity of positive solutions for singular quasilinear problems, J. Math. Anal. Appl. 323 (2006), 12381252.Google Scholar
10.Perera, K. and Zhang, Z., Multiple positive solutions of singular p-Laplacian problems by variational methods, Bound. Value Probl. 3 (2005), 377382.Google Scholar
11.Pucci, P. and Serrin, J., A mountain pass theorem, J. Differ. Equ. 60 (1985), 142149.Google Scholar
12.Ricceri, B., A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401410.Google Scholar
13.Ricceri, B., Sublevel sets and global minima of coercive functionals and local minima of their perturbation, J. Nonlinear Convex Anal. 5 (2004), 157168.Google Scholar
14.Ricceri, B., A further three critical points theorem, Nonlinear Anal. 71 (2009), 41514157.Google Scholar
15.Sun, Y., Wu, S. and Long, Y., Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differ. Equ. 176 (2001), 511531.Google Scholar
16.Vázquez, J. L., A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191202.Google Scholar
17.Zhang, Z., Critical points and positive solutions of singular elliptic boundary value problems, J. Math. Anal. Appl. 302 (2005), 476–83.Google Scholar
18.Zhao, L., He, Y. and Zhao, P., The existence of three positive solutions of a singular p-Laplacian problem, Nonlinear Anal. 74 (2011), 57455753.Google Scholar