Principal eigenvalues for the fractional p-Laplacian with unbounded sign-changing weights
DOI:
https://doi.org/10.58997/ejde.2023.38Keywords:
Fractional p-Laplacian; fractional Sobolev space; indefinite weight; principal eigenvaluesAbstract
Let \(\Omega\) be a bounded regular domain of \( \mathbb{R}^N\), \(N\geqslant 1\), \(p\in (1,+\infty)\), and \( s\in (0,1) \). We consider the eigenvalue problem $$\displaylines{ (-\Delta_p)^s u + V|u|^{p-2}u= \lambda m(x)|u|^{p-2}u \quad\hbox{in } \Omega \cr u=0 \quad \hbox{in } \mathbb{R}^N \setminus \Omega, }$$ where the potential V and the weight m are possibly unbounded and are sign-changing. After establishing the boundedness and regularity of weak solutions, we prove that this problem admits principal eigenvalues under certain conditions. We also show that when such eigenvalues exist, they are simple and isolated in the spectrum of the operator.
For more information see https://ejde.math.txstate.edu/Volumes/2023/38/abstr.html
References
S. Amgibech; On the discrete version of Picone's identity, Discrete and applied Mathematics, 156 (2008), no. 1, 1-10.
P. A. Binding, Y. X. Huang; The principal eigencurve for the p-Laplacian, Di erential and Integral equations, 8 (1995), no. 2, 405-414.
P. A. Binding, Y. X. Huang; Existence and nonexistence of positive eigenfunctions for the p-Laplacian, Procc. American Math. Society, 123 (1995), no. 6, 1833-1838.
L. Brasco, E. Lindgren, E. Parini; The fractional cheeger problem, Interfaces and Free Boundaries 16 (2014), no. 3, 419-458.
L. Brasco, E. Parini, M. Squassina; Stability of variational eigenvalues for the fractional p-Laplacian, Discrete and Contin. Dyn. Sys., 36 (2016), no. 4, 1813-1845.
L. Brasco, E. Parini; The second eigenvalue of the fractional p-Laplacian, Advances in Calculus of Variations, 9 (2016), no. 4, 323-355.
J. Bourgain, H. Br ezis, P. Minorescu; Another look at Sobolev spaces. Optimal Control and PDE (Conference paris 2000 in honour of Prof. Alain Bensoussans's 60th birthday. Eds. J. L. Menaldi and others. Amsterdam IOS Press 2001, 439-455.
M. Cuesta, R. Q. Humberto; A weighted eigenvalue problem for the p-Laplacian plus a potential, NoDEA-Nonlinear di erential equations and applications, 16 (2009), no. 4, 469-491.
M. Cuesta, L. Leadi; Weighted eigenvalue problems for quasilinear elliptic operators with mixed Robin-Dirichlet boundary conditions, J. Math. Anal. Appl., 422 (2015), no. 1, 1-26.
L. M. Del Pezzo, A. Quaas; Global bifurcation for fractional p-Laplacian and application, Z. Anal. Anwend., 35 (2016), no. 4, 411-447.
L. M. Del Pezzo, A. Quaas; Non-resonant Fredholm alternative and anti-maximum principle for the fractional p-Laplacian, J. Fixed Point Theory Appl., 19 (2017), no. 1, 939-958.
L. M. Del Pezzo, J. Fern andez Bonder, L. Lopez Rios; An optimization problem for the first eigenvalue of the p-fractional Laplacian, Mathematische Nachrichten, 291 (2018), no. 4, 632-651.
L. M. Del Pezzo, J. D. Rossi; Eigenvalues for a nonlocal pseudo p-Laplacian, Discrete and continuous Dynamical Systems, 36 (2016), no. 12, 6737-6765.
F. Demengel, G. Demengel; Functional spaces for the theory of elliptic partial di erential equations, Translated from the 2007 French original by Reinie Ern e, Universitext, Springer, London, (2012), EDP Sciences, Les Ulis, 2012, xviii + 465 pp, ISBN: 978-1-4471-2806-9, 978-2-7598-0698-0.
J. Fleckinger, J. Hern andez, F. de Th elin; Existence of Multiple Principal Eigenvalues for some Inde nite Linear Eigenvalue problems, Bollettino della Unione Matematica Italiana Sez.
B Artic. Ric. Mat., (8), 7 (2004), no. 1, 159-188.
G. Franzina, G. Palatucci; Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S), 5 (2014), no. 2, 373-386.
P. Grisvard; Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, (1985), xiv+410 p, ISBN: 0-273-08647-2.
M. Guedda, L. Veron; Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Analysis, Theory, Methods & Applications, 13 (1989), no. 8, 879-902.
P. Hess, T. Kato; On some linear and nonlinear eigenvalue problems with an indefinite weight function. Commun. Partial Di . Equ., 5 (1980), no. 10, 999-1030.
A. Iannizzotto, S. Mosconi, M. Squassina; Global Holder regularity for the fractional p-Laplacian, Revista Matem atic Iberoamericana, 32 (2016), no. 4, 1353-1392. DOI: 10.4171/rmi/921.
L. Leadi, A. Marcos; A weighted eigencurve for Steklov problems with a potential, NoDEA Nonlinear Di erential Equations Appl., 20 (2013), no. 3, 687-713.
R. Servadei, E. Valdinoci; Weak and Viscosity Solutions of the Fractional Laplace Equation, Publ. Mat., 58 (2014), no. 1, 133-154.
T. Kuusi, G. Mingione, Y. Sire; Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), no. 3, 1317-1368.
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