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The existence of positive solutions is considered for the Dirichlet problem\[ \left\{ \begin{array} [c]{rcll}% -\Delta_{p}u & = &\lambda\omega_{1}(x)\left\vert u\right\vert ^{q-2}%u+\beta\omega_{2}(x)\left\vert u\right\vert ^{a-1}u|\nabla u|^{b} & \text{in}\Omega\\ u & = & 0 & \text{on }\partial\Omega, \end{array} \right. \] where$\lambda$ and $\beta$ are positive parameters, $a$ and $b$ are positiveconstants satisfying $a+b\leq p-1$, $\omega_{1}(x)$ and $\omega_{2}(x)$ arenonnegative weights and $1<q\leq p$. The homogeneous case $q=p$ is handled bymaking $q\rightarrow p^{-}$ in the sublinear case $1<q<p,$ which is based onthe sub- and super-solution method. The core of the proof of this problem isthen generalized to the Dirichlet problem $-\Delta_{p}u=f(x,u,\nabla u)$ in$\Omega$, where $f$ is a nonnegative, continuous function satisfying simple,geometrical hypotheses. This approach might be considered as a unification ofarguments dispersed in various papers, with the advantage of handling alsononlinearities that depend on the gradient, even in the $p$-growth case. It isthen applied to the problem $-\Delta_{p}u=\lambda\omega(x)u^{q-1}\left(1+|\nabla u|^{p}\right) $ with Dirichlet boundary conditions in the domain$\Omega$.

A quasilinear problem in two parameters depending on the gradient