Positive solutions for a mixed and singular quasilinear problem

We deal with the existence of solution for the nonlinear elliptic problem {−Δpu=λf(x,u)in Ω,u>0 in Ω,u(x)=0on ∂Ω, where ΩΩ denotes a smooth bounded domain in RNRN, Δpu:=div(∣∇u∣p−2∇u)Δpu:=div(∣∇u∣p−2∇u) with 10λ>0 is a real parameter. The nonlinearity ff is allowed to behave like either f(x,s)⟶s→0∞ and/or f(x,s)⟶s→∞∞ for each x∈Ωx∈Ω.