## On the strict monotonicity of the first eigenvalue of the $p$-Laplacian on annuli

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- by T. V. Anoop, Vladimir Bobkov and Sarath Sasi PDF
- Trans. Amer. Math. Soc.
**370**(2018), 7181-7199 Request permission

## Abstract:

Let $B_1$ be a ball in $\mathbb {R}^N$ centred at the origin and let $B_0$ be a smaller ball compactly contained in $B_1$. For $p\in (1, \infty )$, using the shape derivative method, we show that the first eigenvalue of the $p$-Laplacian in annulus $B_1\setminus \overline {B_0}$ strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as $p \to 1$ and $p \to \infty$ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the $p$-Laplacian on bounded radial domains.## References

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## Additional Information

**T. V. Anoop**- Affiliation: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
- Email: anoop@iitm.ac.in
**Vladimir Bobkov**- Affiliation: Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, Plzeň 306 14, Czech Republic — and — Institute of Mathematics, Ufa Scientific Center, Russian Academy of Sciences, Chernyshevsky str. 112, Ufa 450008, Russia
- MR Author ID: 1040393
- Email: bobkov@kma.zcu.cz
**Sarath Sasi**- Affiliation: School of Mathematical Sciences, National Institute of Science Education and Research Bhubaneswar, HBNI, Jatni 752050, India
- Address at time of publication: Indian Institute of Technology Palakkad, Ahalia Integrated Campus, Kozhipara, Palakkad 678557, Kerala, India
- Email: sarath@iitpkd.ac.in
- Received by editor(s): November 10, 2016
- Received by editor(s) in revised form: March 15, 2017
- Published electronically: June 26, 2018
- Additional Notes: The second author was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 7181-7199 - MSC (2010): Primary 35J92, 35P30, 35B06, 49R05
- DOI: https://doi.org/10.1090/tran/7241
- MathSciNet review: 3841846