Multiple solutions for the pp-Laplace operator with critical growth

In this paper we show the existence of at least three nontrivial solutions to the following quasilinear elliptic equation −Δpu=|u|p∗−2u+λf(x,u)−Δpu=|u|p∗−2u+λf(x,u) in a smooth bounded domain ΩΩ of RNRN with homogeneous Dirichlet boundary conditions on ∂Ω∂Ω, where p∗=Np/(N−p)p∗=Np/(N−p) is the critical Sobolev exponent and Δpu=div(|∇u|p−2∇u)Δpu=div(|∇u|p−2∇u) is the pp-Laplacian. The proof is based on variational arguments and the classical concentration compactness method.