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

AbstractLet us consider the quasilinear problem(Pε){−εpΔpu+up−1=f(u)inΩ,u>0inΩ,u=0on∂Ω where Ω is a bounded domain in RN 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∗[, (Pε) has at least 2P1(Ω)−1 solutions, possibly counted with their multiplicities, where 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)