### Asymptotic behavior of positive solutions of a Dirichlet problem involving supercritical nonlinearities

Let $p>1$, $q>p$, $\lambda >0$ and $s\in ]1,p[$. We study, for $s\to {p}^{-}$, the behavior of positive solutions of the problem $-{\Delta}_{p}u=\lambda {u}^{s-1}+{u}^{q-1}$ in $\Omega $, ${u}_{\mid \partial \Omega}=0$. In particular, we give a positive answer to an open question formulated in a recent paper of the first author.