# zbMATH — the first resource for mathematics

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.
##### MSC:
 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
Full Text:
##### References:
  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  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  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  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  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  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  V. Bobkov and E. Parini, On the higher Cheeger problem, arXiv: 1706.07282. · Zbl 1394.49041  K. Borsuk, Drei Sätze über die $$n$$-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177-190.  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  M. Cuesta, On the Fučík spectrum of the Laplacian and the $$p$$-Laplacian, 2000 Seminar in Differential Equations, May-June 2000, Kvilda (Czech Republic).  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  P. Drábek and S. Robinson, On the generalization of Courant’s Nodal Theorem for the $$p$$-Laplacian, J. Differential Equations 181 (2002), 58-71.  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  A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. · Zbl 1044.55001  B. Helffer and M. Sundqvist, On nodal domains in Euclidean balls, Proc. Amer. Math. Soc. 144 (11), 4777-4791. · Zbl 1362.35200  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  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  P. Juutinen, P. Lindqvist and J.J. Manfredi, The $$∞$$-eigenvalue problem, Arch. Ration. Mech. Anal. 148 (1999), no. 2, 89-105. · Zbl 0947.35104  B. Kawohl and P. Lindqvist, Positive eigenfunctions for the $$p$$-Laplace operator revisited, Analysis (Munich) 26 (2006), no. 4, 545-550. · Zbl 1136.35063  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  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  P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, Providence, 1986. · Zbl 0609.58002  T.M. Trzeciak, Stereographic and cylindrical map projections example, http://www. latex-community.org/viewtopic.php?f=4&t=2111  J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 1, 191-202. · Zbl 0561.35003  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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.