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 : W-0(1,p)(Omega) -> R defined by setting I(u) = 1/p integral(Omega)vertical bar del u vertical bar(p)dx+1/2 integral(Omega)vertical bar del u vertical bar(2)dx- integral F-Omega(x,u)dx, where p> 2, Omega is a bounded domain in R-N, F(x,u) = integral(u)(0) f(x,t)dt and we assume that f is resonant at zero for the spectrum of -Delta in W-0(1,2)(Omega). As an application of these critical groups estimates, some multiplicity results are also given. (C) 2015 Elsevier Inc. All rights reserved.