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}% -Δ_{p}u & = & λh(x,u)+βf(x,u,\nabla u) & \text{in}Ωu & = & 0 & \text{on}\partialΩ, {array}. \hfill] where $h,f$ are continuous nonlinearities satisfying $0\le {\omega }_1(x)u^{q-1}\le h(x,u)\le {\omega }_2(x)u^{q-1}$ with $1<q<p$ and $0\le f(x,u,v)\le {\omega }_3(x)u^a|v{|}^b$, with $a,b>0$, and $\Omega$ is a bounded domain of ${\mathbb{R}}^N,$ $N\ge 3.$ The functions ${\omega }_i$, $1\le i\le 3$, 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.