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 : { − Δ p u = λ h ( x , u ) + β f ( x , u , ∇ u ) in  Ω u = 0 on  ∂ Ω , where h , f are continuous nonlinearities satisfying 0 ≤ ω 1 ( x ) u q − 1 ≤ h ( x , u ) ≤ ω 2 ( x ) u q − 1 with 1 < q < p and 0 ≤ f ( x , u , v ) ≤ ω 3 ( x ) u a | v | b , with a , b > 0 , and Ω is a bounded domain of R N , N ≥ 2 . The functions ω i , 1 ≤ i ≤ 3 , are positive, continuous weights in Ω ¯ . We prove that there exists a region D in the λ β -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.