Morse homology for a class of elliptic partial differential equations

In this paper, we show that a notion of non-degeneracy which allows to develop Morse theory is generically satisfied for a large class of C2-functionals defined on Banach spaces. The main element of novelty with respect to the previous work [L. Asselle and M. Starostka, A note on the Morse homology for a class of functionals in Banach spaces involving the 2p-area functional, NoDEA Nonlinear Differential Equations Appl. 31 (2024) 75, doi:10.1007/s00030-024-00962-3.] of the first and third author is that we do not assume the splitting induced by the second differential at a critical point to persist in a neighborhood, provided one can give precise estimates on how much persistence fails. This allows us to enlarge significantly the class of elliptic pde’s for which non-degeneracy holds and Morse homology can be defined. A concrete example is given by equations involving the p-Laplacian, p ≤ n. As a byproduct, we provide a criterion of independent interest to check whether critical points are non-degenerate in the sense above, and give an abstract construction of Morse homology in a Banach setting for functionals satisfying the Cerami condition.