Critical groups computations on a class of Sobolev Banach spaces via Morse index

In this paper we deal with critical groups estimates for a functional f:W_0^{1,p}(Ω)→R (p>2), Ω bounded domain of R^N, defined by setting f(u)=1/p∫_Ω|∇u|^p dx +1/2∫_Ω|∇u|^2dx+∫_Ω G(u)dx where G(t)=∫_0^t g(s)ds and g is a smooth real function on R, growing subcritically. We remark that the second derivative of f in each critical point u is not a Fredholm operator from W_0^{1,p}(Ω) to its dual space, so that the generalized Morse splitting lemma does not work. In spite of the lack of a Hilbert structure, we compute the critical groups of f in u via its Morse index.