On the Poincaré-Hopf theorem for functionals defined on Banach spaces

Let X be a reflexive Banach space and f a Gâteaux 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 Poincaré-Hopf formula holds. We also show that quasilinear elliptic equations at critical growth are covered by this result.