The Brezis–Nirenberg type problem for the p-laplacian (1 < p < 2): Multiple positive solutions

In this paper we consider the quasilinear critical problem $$(P_λ) ⎨ −Δ_p u = λu^{q-1} + u^{p*-1} in Ω, u&gt;0 in Ω, u =0 on ∂Ω}$$, where Ω is a regular bounded domain in R^N, N≥p^2, 1 &lt;2, p≤q0 is a parameter. In spite of the lack of C^2 regularity of the energy functional associated to (P_λ), we employ new Morse techniques to derive a multiplicity result of solutions. We show that there exists λ*&gt;0 such that, for each λ ∈(0, λ*), either (P_λ) has P_1(Ω) distinct solutions or there exists a sequence of quasilinear problems approximating (P_λ), each of them having at least P_1(Ω) distinct solutions. These results complete those obtained in [23] for the case p≥2.