Multiple positive solutions for a p-Laplace Benci-Cerami type problem (1 < p < 2), via Morse theory

Let us consider the quasilinear problem(P-epsilon) {-epsilon(p) Delta(p)u + u(p-1) = f(u) in Omega,u > 0 in Omega,u = 0 on theta Omega,where Omega is a bounded domain in R-N with smooth boundary, N >= 2, 1 < p < 2, epsilon > 0 is a parameter and f : R -> R is a continuous function with f(0) = 0, having a subcritical growth. We prove that there exists epsilon(*) > 0 such that, for every epsilon is an element of (0, epsilon(*)), (P-epsilon) has at least 2P(1)(Omega) -1 solutions, possibly counted with their multiplicities, where P-t(Omega) is the Poincare polynomial of Omega. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on Omega, approximating (P-epsilon).