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Topological Methods in Nonlinear Analysis

On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian
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On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian

Authors

  • Benjamin Audoux
  • Vladimir Bobkov
  • Enea Parini

Keywords

$p$-Laplacian, nonlinear eigenvalues, Krasnosel'ski{\u\i} genus, symmetries, multiplicity, degree of map

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'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.

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Published

2018-01-20

How to Cite

1.
AUDOUX, Benjamin, BOBKOV, Vladimir and PARINI, Enea. On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian. Topological Methods in Nonlinear Analysis. Online. 20 January 2018. Vol. 51, no. 2, pp. 565 - 582. [Accessed 4 July 2025].
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