Topological Methods in Nonlinear Analysis

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

Benjamin Audoux, Vladimir Bobkov, and Enea Parini

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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'skiĭ 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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Topol. Methods Nonlinear Anal., Volume 51, Number 2 (2018), 565-582.

First available in Project Euclid: 31 January 2018

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Audoux, Benjamin; Bobkov, Vladimir; Parini, Enea. On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian. Topol. Methods Nonlinear Anal. 51 (2018), no. 2, 565--582. doi:10.12775/TMNA.2017.055.

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