## On the second eigenvalue of combination between local and nonlocal $p$-Laplacian

HTML articles powered by AMS MathViewer

- by Divya Goel and K. Sreenadh PDF
- Proc. Amer. Math. Soc.
**147**(2019), 4315-4327 Request permission

## Abstract:

In this paper, we study the mountain pass characterization of the second eigenvalue of the operator $-\Delta _p u -\Delta _{J,p}u$ and study shape optimization problems related to these eigenvalues.## References

- Antonio Ambrosetti and Paul H. Rabinowitz,
*Dual variational methods in critical point theory and applications*, J. Functional Analysis**14**(1973), 349–381. MR**0370183**, DOI 10.1016/0022-1236(73)90051-7 - Frederick J. Almgren Jr. and Elliott H. Lieb,
*Symmetric decreasing rearrangement is sometimes continuous*, J. Amer. Math. Soc.**2**(1989), no. 4, 683–773. MR**1002633**, DOI 10.1090/S0894-0347-1989-1002633-4 - Fuensanta Andreu-Vaillo, José M. Mazón, Julio D. Rossi, and J. Julián Toledo-Melero,
*Nonlocal diffusion problems*, Mathematical Surveys and Monographs, vol. 165, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. MR**2722295**, DOI 10.1090/surv/165 - F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo,
*The limit as $p\to \infty$ in a nonlocal $p$-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles*, Calc. Var. Partial Differential Equations**35**(2009), no. 3, 279–316. MR**2481827**, DOI 10.1007/s00526-008-0205-2 - F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo,
*A nonlocal $p$-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions*, SIAM J. Math. Anal.**40**(2008/09), no. 5, 1815–1851. MR**2471902**, DOI 10.1137/080720991 - Giovanni Molica Bisci, Vicentiu D. Radulescu, and Raffaella Servadei,
*Variational methods for nonlocal fractional problems*, Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge University Press, Cambridge, 2016. With a foreword by Jean Mawhin. MR**3445279**, DOI 10.1017/CBO9781316282397 - Lorenzo Brasco and Enea Parini,
*The second eigenvalue of the fractional $p$-Laplacian*, Adv. Calc. Var.**9**(2016), no. 4, 323–355. MR**3552458**, DOI 10.1515/acv-2015-0007 - M. Cuesta, D. de Figueiredo, and J.-P. Gossez,
*The beginning of the Fučik spectrum for the $p$-Laplacian*, J. Differential Equations**159**(1999), no. 1, 212–238. MR**1726923**, DOI 10.1006/jdeq.1999.3645 - Rupert L. Frank and Robert Seiringer,
*Non-linear ground state representations and sharp Hardy inequalities*, J. Funct. Anal.**255**(2008), no. 12, 3407–3430. MR**2469027**, DOI 10.1016/j.jfa.2008.05.015 - Giovanni Franzina and Giampiero Palatucci,
*Fractional $p$-eigenvalues*, Riv. Math. Univ. Parma (N.S.)**5**(2014), no. 2, 373–386. MR**3307955** - Divya Goel, Sarika Goyal, and Konijeti Sreenadh,
*First curve of Fučik spectrum for the $p$-fractional Laplacian operator with nonlocal normal boundary conditions*, Electron. J. Differential Equations (2018), Paper No. 74, 21. MR**3831820** - Sarika Goyal,
*On the eigenvalues and Fučik spectrum of $p$-fractional Hardy-Sobolev operator with weight function*, Appl. Anal.**97**(2018), no. 4, 633–658. MR**3772129**, DOI 10.1080/00036811.2017.1281406 - Antoine Henrot,
*Extremum problems for eigenvalues of elliptic operators*, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. MR**2251558** - Erik Lindgren and Peter Lindqvist,
*Fractional eigenvalues*, Calc. Var. Partial Differential Equations**49**(2014), no. 1-2, 795–826. MR**3148135**, DOI 10.1007/s00526-013-0600-1 - Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci,
*Hitchhiker’s guide to the fractional Sobolev spaces*, Bull. Sci. Math.**136**(2012), no. 5, 521–573. MR**2944369**, DOI 10.1016/j.bulsci.2011.12.004 - L. M. Del Pezzo, R. Ferreira, and J. D. Rossi,
*Eigenvalues for a combination between local and nonlocal $p$-Laplacians*, arXiv:1803.07988. - K. Sreenadh,
*On the second eigenvalue of a Hardy-Sobolev operator*, Electron. J. Differential Equations (2004), No. 12, 9. MR**2036196**

## Additional Information

**Divya Goel**- Affiliation: Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi-110016, India
- MR Author ID: 1279513
- Email: divyagoel2511@gmail.com
**K. Sreenadh**- Affiliation: Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi-110016, India
- MR Author ID: 693100
- Email: sreenadh@maths.iitd.ac.in
- Received by editor(s): August 30, 2018
- Received by editor(s) in revised form: January 7, 2019
- Published electronically: June 27, 2019
- Communicated by: Catherine Sulem
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 4315-4327 - MSC (2010): Primary 35P30, 49Q10; Secondary 47J10
- DOI: https://doi.org/10.1090/proc/14542
- MathSciNet review: 4002544