Nonlinear elliptic equations having a gradient term with natural growth

In this paper, we study a class of nonlinear elliptic Dirichlet problems whose simplest model example is: {-Delta(p)u = g(u)\del u\(p) + f, in Omega, u = 0, on partial derivative Omega. (1) Here Omega is a bounded open set in R-N (N >= 2), Delta(p) denotes the so-called p-Laplace operator (p > 1) and g is a continuous real function. Given f is an element of L-m (Omega) (m > 1), we study under which growth conditions on g problem (1) admits a solution. If m >= N/p, we prove that there exists a solution under assumption (3) (see below), and that it is bounded when m > N/p; while if 1 < m < N/p and g satisfies the condition (4) below, we prove the existence of an unbounded generalized solution. Note that no smallness condition is asked on f. Our methods rely on a priori estimates and compactness arguments and are applied to a large class of equations involving operators of Leray-Lions type. We also make several examples and remarks which give evidence of the optimality of our results. (C) 2005 Elsevier SAS. All rights reserved.