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 article |
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 |
Appears in Collections: | Články / Articles (KMA) OBD |
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