A quasilinear problem with fast growing gradient

In this paper we consider the following Dirichlet problem for the $p$-Laplacian in the positive parameters $λ$ and $β$: [{{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 $0\leqω_{1}(x)u^{q-1}\leq h(x,u)\leqω_{2}(x)u^{q-1}$ with $1<q<p$ and $0\leq f(x,u,v)\leqω_{3}(x)u^{a}|v|^{b}$, with $a,b>0$, and $Ω$ is a bounded domain of $\mathbb{R}^{N},$ $N\geq3.$ The functions $ω_{i}$, $1\leq i\leq3$, are nonnegative, continuous weights in $\barΩ$. We prove that there exists a region $\mathcal{D}$ in the $λβ$-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.

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