Advanced Nonlinear Studies 9 (2009), 679–699 On the Poincar´ e-Hopf Theorem for Functionals Defined on Banach Spaces Silvia * Cingolani Dipartimento di Matematica Politecnico di Bari, 70126 Bari, Italy e-mail: s.cingolani@poliba.it Marco Degiovanni Dipartimento di Matematica e Fisica Universit`a Cattolica del Sacro Cuore, 25121 Brescia, Italy e-mail: m.degiovanni@dmf.unicatt.it Dedicated to Vieri Benci Communicated by Donato Fortunato Abstract Let X be a reflexive Banach space and f : X -→ R a Gˆateaux differentiable function with f ′ demicontinuous and locally of class (S) + . We prove that each isolated critical point of f has critical groups of finite type and that the Poincar´ e- Hopf formula holds. We also show that quasilinear elliptic equations at critical growth are covered by this result. 2000 Mathematics Subject Classification. 58E05, 35J65. Key words. Morse theory, critical groups, Poincar´ e-Hopf theorem, Banach spaces, critical growth problems ∗ The research of the authors was partially supported by the MIUR project “Variational and topological methods in the study of nonlinear phenomena” (PRIN 2007) and by Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (INdAM). 679