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A multiplicity results for a singular quasilinear elliptic equation

Laboratory of Mathematical Analysis and Applications (LMAA-LR11-ES11), Faculty of Mathematical, Physical and Natural Sciences of Tunis, University of Tunis El-Manar, 2092 Tunis, Tunisia

E-mail Address: mounir.hsini@ipeit.rnu.tn

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Kamel Saoudi

https://orcid.org/0000-0002-7647-4814

College of Sciences at Dammam, University of Imam Abdulrahman Bin Faisal, 31441 Dammam, Kingdom of Saudi Arabia

E-mail Address: kmsaoudi@iau.edu.sa

Corresponding author.

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, and

Mouldi Seddik

Laboratory of Mathematical Analysis and Applications (LMAA-LR11-ES11), Faculty of Mathematical, Physical and Natural Sciences of Tunis, University of Tunis El-Manar, 2092 Tunis, Tunisia

E-mail Address: seddikmouldi@gmail.com

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https://doi.org/10.1142/S0129167X21500713Cited by:1 (Source: Crossref)

Abstract

The purpose of this work is to study the following singular problem:

where

$Ω \subset ℝ^{N}$ ,

$N \geq 2$ be a bounded smooth domain,

$λ$ is a positive parameter,

$p \geq 2$ such that

$N \geq p$ and

$0 < α \leq 1$ ,

$p - 1 < q \leq p^{*} - 1,$ where

$p^{*} = \frac{N p}{N - p} .$ We employ the Nehari manifold approach and the fibering maps in order to show the existence of

$T_{q, α}$ such that for all

$λ \in (0, T_{q, α})$ , problem

$(P_{λ})$ has at least two solutions.

Communicated by Shih-Hsien Yu

Keywords:

AMSC: 34B15, 37C25, 35R20

References

1. J. P. Aubin and I. Ekeland , Applied Nonlinear Analysis, Pure Applied Mathematics (Wiley-Interscience Publications, 1984). Google Scholar
2. M. M. Coclite and G. Palmieri , On a singular nonlinear Dirichlet problem, Comm. Partial Differ. Equ. 14(10) (1989) 1315–1327. Crossref, Web of Science, Google Scholar
3. M. G. Crandall, P. H. Rabinowitz and L. Tartar , On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differ. Equ. 2(2) (1977) 193–222. Crossref, Google Scholar
4. Y. Haitao , Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differ. Equ. 189(2) (2003) 487–512. Crossref, Web of Science, Google Scholar
5. J. Giacomoni and K. Saoudi , Multiplicity of positive solutions for a singular and critical problem, Nonlinear Anal. 71(9) (2009) 4060–4077. Crossref, Web of Science, Google Scholar
6. J. Giacomoni, I. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and global multiplicity for a singular and quasilinear equation, Ann. della Scuola Normale Superiore di Pisa, Classe di Sci. série V, 6 (2007) 117–158. Google Scholar
7. K. Saoudi , Existence and non-existence of solutions for a singular problem with variable potentials, Electron. J. Differ. Equ. 2017(291) (2017) 1–9. Google Scholar
8. K. Saoudi and M. Kratou , Existence of multiple solutions for a singular and quasilinear equation, Complex Var. Elliptic Equ. 60(7) (2015) 893–925. Crossref, Web of Science, Google Scholar
9. Y. J. Sun and S. P. Wu , An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal. 260(5) (2011) 1257–1284. Crossref, Web of Science, Google Scholar
10. S. Yijing, W. Shaoping and L. Yiming , Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differ. Equ. 176(2) (2001) 511–531. Crossref, Web of Science, Google Scholar