Multiplicity Results for p-Laplacian with Critical Nonlinearity of Concave-Convex Type and Sign-Changing Weight Functions

Abstract

The multiple results of positive solutions for the following quasilinear elliptic equation: −Δ𝑝𝑢=𝜆𝑓(𝑥)|𝑢|𝑞−2𝑢+𝑔(𝑥)|𝑢|𝑝∗−2𝑢 in Ω,𝑢=0 on 𝜕Ω, are established. Here, 0∈Ω is a bounded smooth domain in ℝ𝑁,Δ𝑝 denotes the 𝑝-Laplacian operator, 1≤𝑞<𝑝<𝑁,𝑝∗=𝑁𝑝/(𝑁−𝑝),𝜆 is a positive real parameter, and 𝑓,𝑔 are continuous functions on Ω which are somewhere positive but which may change sign on Ω. The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.