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 /g=W//m=subseteq/ R-N. By means of topological arguments, we show how symmetries of /g=W/ help to construct subsets of W-0(1,p) (/g=W/) with suitably high Krasnosel'skii genus. In particular, if /g=W/ is a ball B /m=subseteq/ R-N, we obtain the following chain of inequalities: /g=l/(2)(p;B) /m=le/ ... /m=le//g=l/(N+1)(p;B) /m=le/ /g=l//g=Q/(p; B). Here /g=l/(i)(p; B) are variational eigenvalues of the p-Laplacian on B, and /g=l//g=Q/(p;B) is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of B. If /g=l/(2) (p; B) = /g=l//g=Q/(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 = /m=infty/ are also considered.