A quasilinear problem with fast growing gradient

In this paper we consider the following Dirichlet problem for the $p$-Laplacian in the positive parameters $\lambda$ and $\beta$: [{{array} [c]{rcll}% -\Delta_{p}u & = & \lambda h(x,u)+\beta f(x,u,\nabla u) & $\text{in}$\Omega u & = & 0 & $\text{on}$\partial\Omega, {array}. \hfill] where $h,f$ are continuous nonlinearities satisfying $0\leq\omega_{1}(x)u^{q-1}\leq h(x,u)\leq\omega_{2}(x)u^{q-1}$with$1<q<p$and$0\leq f(x,u,v)\leq\omega_{3}(x)u^{a}|v|^{b}$,with$a,b>0$,and$\Omega$ is a bounded domain of ${\mathbb{R}}^N,$ $N\ge 3.$ The functions ${\omega }_i$, $1\leq i\leq3$,arenonnegative,continuousweightsin$\bar{\Omega}$. We prove that there exists a region $\mathcal{D}$ in the $\lambda \beta$-plane where the Dirichlet problem has at least one positive solution. The novelty in this paper is that our result is valid for nonlinearities with growth higher than $p$ in the gradient variable.