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On some unexpected properties of radial and symmetric eigenvalues and eigenfunctions of the \(p\)-Laplacian on a disk. (English) Zbl 1366.35109
Authors’ abstract: We discuss several properties of eigenvalues and eigenfunctions of the \(p\)-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among the main results, in two dimensions, we show the existence of nonradial eigenfunctions which correspond to the radial eigenvalues. Also, we prove the existence of eigenfunctions whose nodal sets take shapes that cannot occur in the linear case \(p=2\). Moreover, the limit behavior of some eigenvalues as \(p\to +1\) and \(p\to +\infty\) is studied.
Reviewer’s remarks: In the present paper, variational, radial and symmetric eigenvalues and corresponding eigenfunctions of the \(p\)-Laplacian on a ball are characterized in a precise manner, in comparison to the linear case \(p=2\). It may be regarded in part as a continuation of the very recent work of T. V. Anoop et al. [Proc. Am. Math. Soc. 144, No. 6, 2503–2512 (2016; Zbl 1386.35288)].

MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35P15 Estimates of eigenvalues in context of PDEs
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
49R05 Variational methods for eigenvalues of operators
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References:
[1] Anane, A., Simplicité et isolation de la premiere valeur propre du p-laplacien avec poids, C. R. Math. Acad. Sci. Paris, 305, 16, 725-728, (1987) · Zbl 0633.35061
[2] Allegretto, W.; Huang, Y., A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal., 32, 7, 819-830, (1998) · Zbl 0930.35053
[3] Anoop, T. V.; Drábek, P.; Sasi, S., On the structure of the second eigenfunctions of the p-Laplacian on a ball, Proc. Amer. Math. Soc., 144, 6, 2503-2512, (2016) · Zbl 1386.35288
[4] Benedikt, J.; Drábek, P.; Girg, P., The second eigenfunction of the p-Laplacian on the disk is not radial, Nonlinear Anal., 75, 12, 4422-4435, (2012) · Zbl 1251.35057
[5] Brasco, L.; Nitsch, C.; Trombetti, C., An inequality à la Szegő-weinberger for the p-Laplacian on convex sets, Commun. Contemp. Math., 18, 06, (2016) · Zbl 1348.35160
[6] Chorwadwala, A. M.; Mahadevan, R., An eigenvalue optimization problem for the p-Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 145, 06, 1145-1151, (2015) · Zbl 1338.49086
[7] Cuesta, M.; De Figueiredo, D. G.; Gossez, J. P., A nodal domain property for the p-Laplacian, C. R. Math. Acad. Sci. Paris, 330, 8, 669-673, (2000) · Zbl 0954.35124
[8] Del Pino, M. A.; Manasevich, R. F., Global bifurcation from the eigenvalues of the p-Laplacian, J. Differential Equations, 92, 2, 226-251, (1991) · Zbl 0781.35017
[9] Drábek, P.; Robinson, S. B., Resonance problems for the p-Laplacian, J. Funct. Anal., 169, 1, 189-200, (1999) · Zbl 0940.35087
[10] Esposito, L.; Ferone, V.; Kawohl, B.; Nitsch, C.; Trombetti, C., The longest shortest fence and sharp Poincaré-Sobolev inequalities, Arch. Ration. Mech. Anal., 206, 3, 821-851, (2012) · Zbl 1262.52001
[11] García Azorero, J. P.; Peral Alonso, I., Existence and nonuniqueness for the p-Laplacian, Comm. Partial Differential Equations, 12, 12, 126-202, (1987) · Zbl 0637.35069
[12] García Melián, J.; Sabina de Lis, J., On the perturbation of eigenvalues for the p-Laplacian, C. R. Math. Acad. Sci. Paris, 332, 10, 893-898, (2001) · Zbl 0989.35103
[13] Horak, J., Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains, Electron. J. Differential Equations, 2011, 132, 1-30, (2011) · Zbl 1228.35159
[14] Huang, Y., On the eigenvalues of the p-Laplacian with varying p, Proc. Amer. Math. Soc., 125, 11, 3347-3354, (1997) · Zbl 0882.35087
[15] Ifantis, E. K.; Siafarikas, P. D., A differential equation for the zeros of Bessel functions, Appl. Anal., 20, 3-4, 269-281, (1985) · Zbl 0553.33004
[16] Juutinen, P.; Lindqvist, P., On the higher eigenvalues for the ∞-eigenvalue problem, Calc. Var. Partial Differential Equations, 23, 2, 169-192, (2005) · Zbl 1080.35057
[17] Juutinen, P.; Lindqvist, P.; Manfredi, J. J., The ∞-eigenvalue problem, Arch. Ration. Mech. Anal., 148, 2, 89-105, (1999) · Zbl 0947.35104
[18] Kawohl, B.; Fridman, V., Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolin., 44, 4, 659-667, (2003) · Zbl 1105.35029
[19] Kawohl, B.; Lachand-Robert, T., Characterization of Cheeger sets for convex subsets of the plane, Pacific J. Math., 225, 1, 103-118, (2006) · Zbl 1133.52002
[20] Krejčiřik, D.; Pratelli, A., The Cheeger constant of curved strips, Pacific J. Math., 254, 2, 309-333, (2012) · Zbl 1247.28003
[21] Lefton, L.; Wei, D., Numerical approximation of the first eigenpair of the p-Laplacian using finite elements and the penalty method, Numer. Funct. Anal. Optim., 18, 3-4, 389-399, (1997) · Zbl 0884.65103
[22] Niven, I. M., Irrational numbers, (1956), Wiley New York · Zbl 0070.27101
[23] Parini, E., Continuity of the variational eigenvalues of the p-Laplacian with respect to p, Bull. Aust. Math. Soc., 83, 03, 376-381, (2011) · Zbl 1217.35130
[24] Parini, E., The second eigenvalue of the p-Laplacian as p goes to 1, Int. J. Differ. Equ., 2010, 1-23, (2010) · Zbl 1207.35235
[25] Perera, K., Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal., 21, 2, 301-309, (2003) · Zbl 1039.47041
[26] Watson, G. N., A treatise on the theory of Bessel functions, (1944), The University Press Cambridge · Zbl 0063.08184
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