We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with Singular lower order terms that have natural growth with respect to the gradient, whose model is {-Delta u + vertical bar del u vertical bar(2)/u(gamma) = f in Omega. u = 0 on partial derivative Omega. where Omega is an open bounded subset of R, gamma > 0 and f is a function which is strictly positive on every compactly contained subset of Omega. As a consequence of our main results, we prove that the condition gamma < 2 is necessary and sufficient for the existence of solutions in H(0)(1) (Omega) for every sufficiently regular f as above. (C) 2009 Elsevier Inc. All rights reserved

A semilinear elliptic equation with a mild singularity at u = 0:existence and homogenization ------------revised version, March 8, 2016 accepted for publication in J. Math. Pures et Appl. Daniela Giachetti AbstractIn this paper we consider singular semilinear elliptic equations whose prototype is the following

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Given a bounded domain Ω in R N , and a function a ∈ L q (Ω) with q > N/2, we study the existence of a positive solution for the quasilinear problemwhere g(x, s) is a Carathéodory function on Ω × (0, +∞) which may have a singularity at s = 0 and may change of sign.

We study the existence of positive solution w ∈ H 1 0 (Ω) of the quasilinear equation −∆w + g(w)|∇w| 2 = a(x), x ∈ Ω, where Ω is a bounded domain in R N , 0 ≤ a ∈ L ∞ (Ω) and g is a nonnegative continuous function on (0, +∞) which may have a singularity at zero.

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