A note on the Morse homology for a class of functionals in Banach spaces involving the 2p-area functional

In this paper we show how to construct Morse homology for an explicit class of functionals involving the 2p-area functional. 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, in the case \(p>n/2\), the results in Cingolani and Vannella (Ann Inst H Poincaré Anal Non Linéaire 2:271–292, 2003; Ann Mat Pura Appl 186:155–183, 2007), 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