Název:	On multiplicity of eigenvalues and symmetry of eigenfunctions of the p--Laplacian
Autoři:	Audoux, Benjamin Bobkov, Vladimír Parini, Enea
Datum vydání:	2018
Nakladatel:	Nicolaus Copernicus University in Torun, Juliusz Schauder Centre for Nonlinear Studies
Typ dokumentu:	článek article
URI:	http://hdl.handle.net/11025/30447
ISSN:	1230-3429
Klíčová slova v dalším jazyce:	p-Laplacian;nonlinear eigenvalues;Krasnoselskii genus;symmetry;multiplicity;degree of map.
Abstrakt v dalším jazyce:	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.
Práva:	Plný text není přístupný. © Nicolaus Copernicus University in Torun, Juliusz Schauder Centre for Nonlinear Studies
Vyskytuje se v kolekcích:	Články / Articles (KMA) OBD

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