A quasilinear problem with fast growing gradient

In this paper, we consider the following Dirichlet problem for the pp-Laplacian in the positive parameters λλ and ββ: {−Δpu=λh(x,u)+βf(x,u,∇u)in Ωu=0on ∂Ω,where h,fh,f are continuous nonlinearities satisfying 0≤ω1(x)uq−1≤h(x,u)≤ω2(x)uq−10≤ω1(x)uq−1≤h(x,u)≤ω2(x)uq−1 with 10a,b>0, and ΩΩ is a bounded domain of RN,N≥2RN,N≥2. The functions ωiωi, 1≤i≤31≤i≤3, are positive, continuous weights in Ω¯. We prove that there exists a region DD in the λβλβ-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 pp in the gradient variable.