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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.