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

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 Krasnoselskiu{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.