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

Let us consider the quasilinear problem(Pε){−εpΔpu+up−1=f(u)inΩ,u>0inΩ,u=0on∂Ω where Ω   is a bounded domain in RNRN with smooth boundary, N>pN>p, 2⩽p0ε>0 is a parameter. We prove that there exists ε∗>0ε∗>0 such that, for any ε∈]0,ε∗[ε∈]0,ε∗[, (Pε)(Pε) has at least 2P1(Ω)−12P1(Ω)−1 solutions, possibly counted with their multiplicities, where Pt(Ω)Pt(Ω) 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)(Pε).