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.
Cingolani, S., Degiovanni, M., On the Poincaré-Hopf theorem for functionals defined on Banach spaces, <<ADVANCED NONLINEAR STUDIES>>, 2009; 9 (4): 679-699. [doi:10.1515/ans-2009-0406] [http://hdl.handle.net/10807/3483]
On the Poincaré-Hopf theorem for functionals defined on Banach spaces
Degiovanni, Marco
2009
Abstract
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.
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