Summary: Let \(p>1\), \(q>p\), \(\lambda >0\) and \(s\in ]1,p[\). We study, for \(s\rightarrow p^-\), 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.