Giovanni Franzina^[1] and Giampiero Palatucci^[2]

Fractional p-eigenvalues

Pages: 373-386
Received: 10 June 2013   
Accepted in revised form: 5 July 2013
Mathematics Subject Classification (2010): 35J60, 35P30, 35R11.

Keywords: Nonlinear eigenvalues problems, nonlocal problem, fractional Laplacian, quasilinear nonlocal operators, Dirichlet forms, Caccioppoli estimates.
Authors addresses:
[1] : Departiment Mathematik, Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
[2] : Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Campus - Parco Area delle Scienze, 53/A, 43124 Parma, Italia

Abstract: We discuss some basic properties of the eigenfunctions of a class of nonlocal operators whose model is the fractional p-Laplacian.

The first author has been supported by the ERC grant 258685 "AnOptSetCon''. The second author has been supported by the ERC grant 207573 "Vectorial Problems''.

References

[1] M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the p-Laplace operator, Manuscripta Math. 109 (2002), no. 2, 229-231. [MR1935031]
[2] M. Belloni and B. Kawohl, The pseudo-p-Laplace eigenvalue problem and viscosity solutions as \(p\to\infty\), ESAIM Control Optim. Calc. Var. 10 (2004), no. 1, 28-52.[MR2084254]
[3] R. Benguria, H. Brézis and E. H. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys. 79 (1981), no. 2, 167-180. [MR0612246]
[4] L. Brasco and G. Franzina, A note on positive eigenfunctions and hidden convexity, Arch. Math. (Basel) 99 (2012), no. 4, 367-374. [MR2990155]
[5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. [MR2354493]
[6] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, submitted paper.
[7] A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), no. 6, 1807-1836. [MR3237774]
[8] A. Di Castro and G. Palatucci, Fractional regularity for nonlinear elliptic problems with measure data, J. Convex Anal. 20 (2013), no. 4, 901-918. [MR3184287]
[9] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521-573. [MR2944369]
[10] G. Franzina and E. Valdinoci, Geometric analysis of fractional phase transition interfaces. Geometric properties for parabolic and elliptic PDE's, Springer INdAM Ser., 2, Springer, Milan 2013, 117-130. [MR3050230]
[11] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations 34 (2009), no. 1, 1-21. [MR2448308]
[12] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London 1968. [MR0244627]
[13] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 795-826. [MR3148135]
[14] P. Lindqvist, On the equation \({\rm div} (|\nabla u|^{p-2}\nabla u) +\lambda |u|^{p-2}u=0\), Proc. Amer. Math. Soc. 109 (1990), no. 1, 157-164. [MR1007505]
[15] P. Lindqvist, Addendum: "On the equation \({\rm div}(\vert \nabla u\vert ^{p-2}\nabla u)+\lambda\vert u\vert ^{p-2}u=0\)" , Proc. Amer. Math. Soc. 116 (1992), no. 2, 583-584. [MR1139483]
[16] G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 2, 195-261. [MR2352517]
[17] G. Mingione, Gradient potential estimates, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 2, 459-486. [MR2746772]
[18] G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations 50 (2014), no. 3-4, 799-829. [MR3216834]
[19] G. Palatucci, A. Pisante and Y. Sire, Subcritical approximation of a Yamabe type non local equation: a Gamma-convergence approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5). [DOI]
[20] G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl. (4) 192 (2013), no. 4, 673-718. [MR3081641]
[21] O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, J. Math. Pures Appl. (9) 101 (2014), no. 1, 1-26. [MR3133422]
[22] R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal. 12 (2013), no. 6, 2445- 2464. [MR3060890]
[23] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 2105-2137. [MR3002745]
[24] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat. 58 (2014), no. 1, 133-154. [MR3161511]