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

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\uı genus. In particular, if $\Omega$ is a ball $B\subset {\mathbb{R}}^N$, we obtain the following chain of inequalities: $${\lambda }_2(p;B)\le \cdots \le {\lambda }_{N+1}(p;B)\le {\lambda }_{\ominus }(p;B).$$ 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.

arXiv comment14 pages, 1 figure