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

Let us consider the quasilinear problem(P_ε) {(-ε^p Δ_p u + u^{p-1} =f(u), in Ω,; u>0, in Ω,; u=0, on ∂Ω) where Ω is a bounded domain in R^N with smooth boundary, N > p, 2 ≤ p < p*, p* = Np /(N - p), ε > 0 is a parameter. We prove that there exists ε* > 0 such that, for any ε ∈]0,ε*[, (P_ε) has at least 2P_1(Ω)-1 solutions, possibly counted with their multiplicities, where P_t(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 (P_ε).