Description

• Title:
  On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian
• Creator:
  Audoux, Benjamin;
  Bobkov, Vladimir;
  Parini, Enea
• Description:
  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'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.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2018 Topological Methods in Nonlinear Analysis
• Date:
  2018.01.20
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 51, No 2 (June 2018); 565 - 582;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2017.055/13994 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2017.055/13995
• Subject:
  $p$-Laplacian;
  nonlinear eigenvalues;
  Krasnosel'ski{\u\i} genus;
  symmetries;
  multiplicity;
  degree of map;
  Primary;
  35J92;
  35P30;
  35B06;
  Secondary;
  35A15;
  35A16;
  55M25
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2017.055
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other