Title: On multiplicity of eigenvalues and symmetry of eigenfunctions of the p--Laplacian
Authors: Audoux, Benjamin
Bobkov, Vladimír
Parini, Enea
Issue Date: 2018
Publisher: Nicolaus Copernicus University in Torun, Juliusz Schauder Centre for Nonlinear Studies
Document type: článek
URI: http://hdl.handle.net/11025/30447
ISSN: 1230-3429
Keywords in different language: p-Laplacian;nonlinear eigenvalues;Krasnoselskii genus;symmetry;multiplicity;degree of map.
Abstract in different language: We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $\Omega \subset \R^N$. 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'ski\u{\i} 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.
Rights: Plný text není přístupný.
© Nicolaus Copernicus University in Torun, Juliusz Schauder Centre for Nonlinear Studies
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