  We discuss several properties of eigenvalues and eigenfunctions of the
$p$-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among
main results, in two dimensions, we show the existence of nonradial
eigenfunctions which correspond to the radial eigenvalues. Also we prove the
existence of eigenfunctions whose shape of the nodal set cannot occur in the
linear case $p=2$. Moreover, the limit behavior of some eigenvalues as $p \to
1+$ and $p \to +\infty$ is studied.
