The Brézis–Nirenberg type problem for the p -Laplacian: infinitely many positive solutions

In this paper, we consider the quasilinear critical problem {−Δpu=λ|u|p−2u+|u|p⋆−2uinΩ,u=0on∂Ω, where Ω is a bounded domain in RN with smooth boundary, −Δpu:=div(|∇u|p−2∇u) is the p-Laplacian, N>p,1 0 is a parameter. We show that when the parameter λ>0 is slighty smaller than λ1(Ω) (λ1(Ω) being the first eigenvalue of −Δp in Ω under Dirichlet boundary condition), the problem admits infinitely many positive solutions. These results complete those obtained in Cingolani and Vannella (J. Differential Equations 266 (2019) 4510–4532) for the case of P1(Ω) positive solutions (P1(Ω) being the Poincaré polynomial of Ω), in Azorero and Peral (Comm. Partial Differential Equations 12 (1987) 1389–1430) for the case of one positive solution, and in Mercuri and Perera (J. Funct. Anal. 283 (2022) 109536) for the case of N/2 distinct pairs of nontrivial solutions. Our approach uses variational methods together with upper-lower solutions and suitable truncation techniques, and flow invariance arguments.