Uniqueness of positive solutions for a class of quasilinear problems

We prove uniqueness of positive solutions for the quasilinear problems −Δpu=λf(u)in Ωu=0on ∂Ω, where Δpu=div(|∇u|p−2∇u),p>1,Ω is a bounded domain in RNRN with smooth boundary ∂Ω,f∂Ω,f is pp-sublinear at ∞,f(u)up−1 is decreasing for large uu, and λλ is a large positive parameter. We also obtain the asymptotic behavior of the solution obtained as λ→∞λ→∞.