On the Dirichlet problem involving the equation −Δpu=λus−1−Δpu=λus−1

We study, with λλ varying in R+R+, the behavior of the LsLs-norm of the unique positive solution of the following Dirichlet problem {−Δpu=λus−1in Ωu∣∂Ω=0 when s→p−s→p−. In particular, we prove that this norm is asymptotic to C(λλp)1p−s for some positive constant CC, where λpλp is the first eigenvalue of the pp-Laplacian on ΩΩ.