  In this paper we show how to construct Morse homology for an explicit class
of functionals involving the $p$-Laplacian. The natural domain of definition of
such functionals is the Banach space $W^{1,2p}_0(\Omega)$, where $p>n/2$ and
$\Omega \subset \mathbb R^n$ is a bounded domain with sufficiently smooth
boundary. As $W^{1,2p}_0(\Omega)$ is not isomorphic to its dual space, critical
points of such functionals cannot be non-degenerate in the usual sense, and
hence in the construction of Morse homology we only require that the second
differential at each critical point be injective. Our result upgrades a result
of Cingolani and Vannella, where critical groups for an analogous class of
functionals are computed, and provides in this special case a positive answer
to Smale's suggestion that injectivity of the second differential should be
enough for Morse theory.
