Multiple positive solutions for a critical quasilinear equation via Morse theory

We deal with the existence of solutions for the quasilinear problem(P_λ) {(- Δ_p u = λ u^{q-1} + u^{p*-1}, 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, 1<p≤q<p*, p*=Np/(N-p), λ>0 is a parameter. Using Morse techniques in a Banach setting, we prove that there exists λ* > 0 such that, for any λ ∈ (0, λ*), (P_λ) has at least P_1(Ω) solutions, possibly counted with their multiplicities, where P_t (Ω) is the Poincaré polynomial of Ω. Moreover for p ≥ 2 we prove that, for each λ ∈ (0, λ*), there exists a sequence of quasilinear problems, approximating (P_λ), each of them having at least P_1(Ω) distinct positive solutions.