{{\small The aim of this paper is investigating the existence of solutions of the quasilinear elliptic problem $$ \leqno{(P_\varphi)}\qquad\qquad \left\{ \begin{array}{ll} \displaystyle{-\Delta_p u\ =|u|^{q-2}u + f} & \mbox{ in } \Omega,\\ \displaystyle{u=\varphi} & \mbox{ on } \partial\Omega,\\ \end{array} \right. $$ where $\Omega$ is an open bounded domain of ${\bf R}^N$ with $C^2$ boundary $\partial\Omega$, $\Delta_p u= {\rm div}(|\nabla u|^{p-2}\nabla u)$, $1<p<q<p^\ast$, $f\in C(\overline\Omega)$ and $\varphi\in C^2(\overline\Omega)$. By means of the so-called Bolle's method in \cite{bolle, bgt}, we extend a result in \cite{gp} where the authors consider $\Omega = ]0,1[^N$ and $u=0$ on $\partial\Omega$. }} \end{center}