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On multiplicity of eigenvalues and symmetry of eigenfunctions of the \(p\)-Laplacian. (English) Zbl 1410.35050
Authors’ abstract: We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the \(p\)-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains \(\Omega \subset \mathbb{R}^N.\) By means of topological arguments, we show how symmetries of \(\Omega\) help to construct subsets of \(W_0^{1,p}(\Omega)\) with suitably high Krasnosel’skiĭ genus. In particular, if \(\Omega\) is a ball \(B \subset \mathbb{R}^N,\) we obtain the following chain of inequalities: \[ \lambda_2(p;B) \leq \dots \leq \lambda_{N+1}(p;B) \leq \lambda_\ominus(p;B). \] Here \(\lambda_i(p;B)\) are variational eigenvalues of the \(p\)-Laplacian on \(B\), and \(\lambda_\ominus(p;B)\) is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of \(B\). If \(\lambda_2(p;B) = \lambda_\ominus(p;B)\), as it holds true for \(p=2\), the result implies that the multiplicity of the second eigenvalue is at least \(N\). In the case \(N=2\), we can deduce that any third eigenfunction of the \(p\)-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases \(p=1\), \(p=\infty\) are also considered.
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
35A15 Variational methods applied to PDEs
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[1] A. Anane, Simplicité et isolation de la première valeur propre du \(p\)-Laplacien avec poids, C.R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 16, 725-728. · Zbl 0633.35061
[2] T.V. Anoop, P. Drábek and S. Sasi, On the structure of the second eigenfunctions of the \(p\)-Laplacian on a ball, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2503-2512. · Zbl 1386.35288
[3] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and Optimization, second edition, MOS-SIAM Series on Optimization, vol. 17, SIAM, Philadelphia, 2014. · Zbl 1311.49001
[4] M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the \(p\)-Laplace operator, Manuscripta Math. 109 (2002), no. 2, 229-231. · Zbl 1100.35032
[5] J. Benedikt, P. Drábek and P. Girg, The second eigenfunction of the \(p\)-Laplacian on the disc is not radial, Nonlinear Anal. 75 (2012), no. 12, 4422-4435. · Zbl 1251.35057
[6] V. Bobkov and P. Drábek, On some unexpected properties of radial and symmetric eigenvalues and eigenfunctions of the \(p\)-Laplacian on a disk, J. Differential Equations 263 (2017), no. 3, 1755-1772. · Zbl 1366.35109
[7] V. Bobkov and E. Parini, On the higher Cheeger problem, arXiv: 1706.07282. · Zbl 1394.49041
[8] K. Borsuk, Drei Sätze über die \(n\)-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177-190.
[9] L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional \(p\)-Laplacian, Discrete Contin. Dyn. Syst. 36 (2016), no. 4, 1813-1845. · Zbl 1336.35270
[10] M. Cuesta, On the Fučík spectrum of the Laplacian and the \(p\)-Laplacian, 2000 Seminar in Differential Equations, May-June 2000, Kvilda (Czech Republic).
[11] M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the \(p\)-Laplacian, J. Differential Equations 92 (1991), no. 2, 226-251. · Zbl 0781.35017
[12] P. Drábek and S. Robinson, On the generalization of Courant’s Nodal Theorem for the \(p\)-Laplacian, J. Differential Equations 181 (2002), 58-71.
[13] J.P. García Azorero and I. Peral, Existence and nonuniqueness for the \(p\)-Laplacian: nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), no. 12, 1389-1430. · Zbl 0637.35069
[14] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. · Zbl 1044.55001
[15] B. Helffer and M. Sundqvist, On nodal domains in Euclidean balls, Proc. Amer. Math. Soc. 144 (11), 4777-4791. · Zbl 1362.35200
[16] Y. Huang, On the eigenvalues of the \(p\)-Laplacian with varying \(p\), Proc. Amer. Math. Soc. 125 (1997), no. 11, 3347-3354. · Zbl 0882.35087
[17] P. Juutinen and P. Lindqvist, On the higher eigenvalues for the \(∞\)-eigenvalue problem, Calc. Var. Partial Differential Equations 23 (2005), no. 2, 169-192. · Zbl 1080.35057
[18] P. Juutinen, P. Lindqvist and J.J. Manfredi, The \(∞\)-eigenvalue problem, Arch. Ration. Mech. Anal. 148 (1999), no. 2, 89-105. · Zbl 0947.35104
[19] B. Kawohl and P. Lindqvist, Positive eigenfunctions for the \(p\)-Laplace operator revisited, Analysis (Munich) 26 (2006), no. 4, 545-550. · Zbl 1136.35063
[20] S. Littig and F. Schuricht, Convergence of the eigenvalues of the \(p\)-Laplace operator as \(p\) goes to \(1\), Calc. Var. Partial Differential Equations 49 (2014), 707-727. · Zbl 1282.35267
[21] E. Parini, The second eigenvalue of the \(p\)-Laplacian as \(p\) goes to \(1\), Int. J. Differ. Equ. 2010 (2010), Art. ID 984671, 23 pp. · Zbl 1207.35235
[22] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, Providence, 1986. · Zbl 0609.58002
[23] T.M. Trzeciak, Stereographic and cylindrical map projections example, http://www. latex-community.org/viewtopic.php?f=4&t=2111
[24] J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 1, 191-202. · Zbl 0561.35003
[25] X. Yao and J. Zhou, Numerical methods for computing nonlinear eigenpairs. I. Iso-homogeneous cases, SIAM J. Sci. Comput. 29 (2007), no. 4, 1355-1374. · Zbl 1156.65055
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