Critical groups at zero and multiple solutions for a quasilinear elliptic equation

In this paper, by Morse theory we will compute the critical groups at zero for a functional I:W01,p(Ω)→R defined by settingI(u)=1p∫Ω|∇u|pdx+12∫Ω|∇u|2dx−∫ΩF(x,u)dx, where p>2p>2, Ω is a bounded domain in RNRN, F(x,u)=∫0uf(x,t)dt and we assume that f   is resonant at zero for the spectrum of −Δ in W01,2(Ω). As an application of these critical groups estimates, some multiplicity results are also given.