A quasilinear problem with fast growing gradient

Abstract: 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 $\mathbb{R}^{N},$ $N\geq3.$ The functions $\omega_{i}$, $1\leq i\leq3$, are nonnegative, continuous weights in $\bar{\Omega}$. 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.