We prove a result of existence of positive solutions of the Dirichlet problem for $-��_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $��\subset\mathbb{R}^N$, where $��_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight function. As in previous results by the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on $f$, but simple geometric assumptions on a neighborhood of the first eigenvalue of the $p$-Laplacian operator. We start by solving the problem in a radial domain by applying the Schauder Fixed Point Theorem and this result is used to construct an ordered pair of sub- and super-solution, also valid for nonlinearities which are super-linear both at the origin and at $+\infty$. We apply our method to the Dirichlet problem $-��_pu = ��u(x)^{q-1}(1+|\nabla u(x)|^p)$ in $��$ and give examples of super-linear nonlinearities which are also handled by our method.