Multiple positive solutions for a critical quasilinear equation via Morse theory

Abstract

We deal with the existence of solutions for the quasilinear problem(Pλ){−Δpu=λuq−1+up∗−1inΩ,u>0inΩ,u=0on∂Ω, where Ω is a bounded domain in RN with smooth boundary, N⩾p2, 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∗), (Pλ) has at least P1(Ω) solutions, possibly counted with their multiplicities, where Pt(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 (Pλ), each of them having at least P1(Ω) distinct positive solutions.