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

In the present paper we study the behaviour as p goes to 1 of the weak solutions to the problems 8 < :where Ω is a bounded open set of R N (N ≥ 2) with Lipschitz boundary and p > 1. As far as the datum f is concerned, we analyze several cases: the most general one is f ∈ W −1,∞ (Ω). We also illustrate our results by means of remarks and examples. IntroductionIn the present paper we study the behaviour, when p goes to 1, of the solutions u p ∈ W 1,p 0 (Ω) to the problemswhere p > 1 and Ω is a bounded open set of R N (N ≥ 2) with Lipschitz boundary. We analyze the case where Ω is a ball and the datum f is a non-negative radially decreasing function belonging to the Lorentz space L N,∞ (Ω) and the case where the datum f belongs to the dual space W −1,∞ (Ω). We are interested in finding the pointwise limit of u p as p goes to 1 and in proving that such a limit is a solution to the "limit equation" of (1.1), namely:2000 Mathematics Subject Classification. 35J20, 35J70.

Abstract. We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given byThe main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they could be applied to obtain uniqueness results for nonlinear equations having the p-Laplacian operator as the principal part. Our results improve those already known, even if the gradient term is not singular.

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Abstract. In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in L 1 (Ω),where a is a Carathéodory function satisfying the classical Leray-Lions hypothesis, ∂/∂ηa is the Neumann boundary operator associated to a, Du the gradient of u and β is a maximal monotone graph in R × R with 0 ∈ β(0).