A note on the Morse homology for a class of functionals in Banach spaces involving the $p$-Laplacian

TL;DR

This paper develops Morse homology for a class of functionals involving the p-Laplacian in Banach spaces, overcoming degeneracy issues by requiring only injectivity of the second differential at critical points.

Contribution

It constructs Morse homology for p-Laplacian functionals in Banach spaces where the second differential is only injective, extending previous results and confirming Smale's conjecture.

Findings

Morse homology can be constructed with injective second differentials.

The work extends previous critical group computations.

Provides a positive answer to Smale's conjecture in this context.

Abstract

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.