On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$Laplacian
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.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 arXiv:
 arXiv:1704.03194
 Bibcode:
 2017arXiv170403194A
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Spectral Theory;
 35J92;
 35P30;
 35A15;
 35A16;
 55M25;
 35B06
 EPrint:
 14 pages, 1 figure