  The $p$-Laplacian operator $\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla
u\right)$ is not uniformly elliptic for any $p\in(1,2)\cup(2,\infty)$ and
degenerates even more when $p\to \infty$ or $p\to 1$. In those two cases the
Dirichlet and eigenvalue problems associated with the $p$-Laplacian lead to
intriguing geometric questions, because their limits for $p\to\infty$ or $p\to
1$ can be characterized by the geometry of $\Omega$. In this little survey we
recall some well-known results on eigenfunctions of the classical 2-Laplacian
and elaborate on their extensions to general $p\in[1,\infty]$.
  We report also on results concerning the normalized or game-theoretic
$p$-Laplacian $$\Delta_p^Nu:=\tfrac{1}{p}|\nabla
u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$$ and
its parabolic counterpart $u_t-\Delta_p^N u=0$. These equations are homogeneous
of degree 1 and $\Delta_p^N$ is uniformly elliptic for any $p\in (1,\infty)$.
In this respect it is more benign than the $p$-Laplacian, but it is not of
divergence type.
