  We adapt the inverse iteration method for symmetric matrices to some
nonlinear PDE eigenvalue problems. In particular, for $p\in (1,\infty)$ and a
given domain $\Omega\subset\mathbb{R}^n$, we analyze a scheme that allows us to
approximate the smallest value the ratio $\int_\Omega|D\psi|^p
dx/\int_\Omega|\psi|^p dx$ can assume for functions $\psi$ that vanish on
$\partial \Omega$. The scheme in question also provides a natural way to
approximate minimizing $\psi$. Our analysis also extends in the limit as
$p\rightarrow\infty$ and thereby fashions a new approximation method for ground
states of the infinity Laplacian.
