A quasilinear problem with fast growing gradient

In this paper, we consider the following Dirichlet problem for the p-Laplacian in the positive parameters lambda and beta: {-Delta(p)u = lambda h(x, u) + beta f(x, u, del u) in Omega u = 0 on partial derivative Omega, where h, f are continuous nonlinearities satisfying 0 <= omega(1)(x)u(q-1) <= h(x, u) <= omega(2)(x)u(q-1) with 1 < q < p and 0 <= f (x, u, v) <= omega(3)(x)u(a)vertical bar v vertical bar(b), with a, b > 0, and Omega is a bounded domain of R-N, N >= 2. The functions omega(i), 1 <= i <= 3, are positive, continuous weights in (Omega) over bar. We prove that there exists a region 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. (C) 2012 Elsevier Ltd. All rights reserved.