On the multiplicity of positive solutions for p-Laplace equations via Morse theory

Let us consider the quasilinear problem where Ω is a bounded domain in with smooth boundary, N>p, 2 p0 is a parameter. We prove that there exists ε*>0 such that, for any ε ]0,ε*[, (Pε) has at least solutions, possibly counted with their multiplicities, where is the Poincaré polynomial of Ω. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on Ω, approximating (Pε).