## Taiwanese Journal of Mathematics

### Eigenvalue Problem for a System of Singular ODEs with a Perturbed $q$-Laplace operator

#### Abstract

Our purpose is to characterize the eigenvalue interval for a system of boundary value problems with a one-dimensional perturbed $q$-Laplace operator. We consider both sublinear and superlinear nonlinearities with a possible singularity at zero. The tools applied here are based on variational methods and properties of the Fenchel transform.

#### Article information

Source
Taiwanese J. Math., Volume 23, Number 3 (2019), 691-701.

Dates
Received: 22 March 2018
Revised: 25 May 2018
Accepted: 1 October 2018
First available in Project Euclid: 11 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1539223220

Digital Object Identifier
doi:10.11650/tjm/181003

Mathematical Reviews number (MathSciNet)
MR3952247

Zentralblatt MATH identifier
07068570

#### Citation

O'Regan, Donal; Orpel, Aleksandra. Eigenvalue Problem for a System of Singular ODEs with a Perturbed $q$-Laplace operator. Taiwanese J. Math. 23 (2019), no. 3, 691--701. doi:10.11650/tjm/181003. https://projecteuclid.org/euclid.twjm/1539223220

#### References

• D. Bonheure, J. M. Gomes and L. Sanchez, Positive solutions of a second-order singular ordinary differential equation, Nonlinear Anal. 61 (2005), no. 8, 1383–1399.
• I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies in Mathematics and its Application 1, North-Holland, Amsterdam, 1976.
• M. Grossi and D. Passaseo, Nonlinear elliptic Dirichlet problems in exterior domains: the role of geometry and topology of the domain, Comm. Appl. Nonlinear Anal. 2 (1995), no. 2, 1–31.
• E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 1997.
• H. Lü, D. O'Regan and R. P. Agarwal, Existence to singular boundary value problems with sign changing nonlinearities using an approximation method approach, Appl. Math. 52 (2007), no. 2, 117–135.
• ––––, An approximation approach to eigenvalue intervals for singular boundary value problems with sign changing nonlinearities, Math. Inequal. Appl. 11 (2008), no. 1, 81–98.
• R. Molle and D. Passaseo, Multiple solutions of nonlinear elliptic Dirichlet problems in exterior domains, Nonlinear Anal. 39 (2000), no. 4, Ser. A: Theory Methods, 447–462.
• D. O'Regan and A. Orpel, Eigenvalue problems for singular ODEs, Glasg. Math. J. 53 (2011), no. 2, 301–312.
• ––––, Eigenvalue problem for ODEs with a perturbed $q$-Laplace operator, Dynam. Systems Appl. 24 (2015), no. 1-2, 97–112.
• A. Orpel, On the existence of bounded positive solutions for a class of singular BVPs, Nonlinear Anal. 69 (2008), no. 4, 1389–1395.