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

Let us consider the quasilinear problem (P_ε) ⎨−ε^p Δ_p u+u^(p-1)=f(u) in Ω, u&gt;0 in Ω, u=0 on ∂Ω, where Ω is a bounded domain in R^N with smooth boundary, N≥2, 1&lt;2, ε&gt;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 ε^∗&gt;0 such that, for every ε∈(0,ε^∗), (P_ε) has at least 2(P_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).