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We show that, under appropriate structure conditions, the quasilinear Dirichlet problem $$\left\{\begin{array}{ccc}-div\left(\right|\nabla u{|}^{p-2}\nabla u)=f(x,u),\hfill & x\in \Omega ,u=0,\hfill & x\in \partial \Omega ,\end{array}\right.$$ where $\Omega$is a bounded domain in ${ℝ}^n$, $1<p<+\infty$, admits two positive solutions $u_0$, $u_1$ in $W_0^{1,p}\left(\Omega \right)$ such that $0<u_0\le u_1$ in $\Omega$, while $u_0$ is a local minimizer of the associated Euler-Lagrange functional.