Journal of Mathematical Analysis and Applications 263, 522–542 (2001) doi:10.1006/jmaa.2001.7628, available online at http://www.idealibrary.com on Existence and Estimates of Solutions for Singular Nonlinear Elliptic Problems Habib Mˆ aagli and Malek Zribi D´ epartement de Math´ ematiques, Facult´ e des Sciences de Tunis, Campus Universitaire, 1060 Tunis, Tunisia E-mail: habib.maagli@fst.rnu.tn Submitted by C. Eugene Wayne Received October 13, 2000 1. INTRODUCTION In this paper, we shall investigate the existence and the uniqueness of a solution to a nonlinear elliptic problem, u + ϕu= 0 in  in the weak sense u ∂ = 0 (1) for both bounded and unbounded domain  in R d d ≥ 3, with smooth boundary ∂. The function ϕ  ×0 ∞→0 ∞ is measurable and satisfies some appropriate hypotheses detailed below. The problem (1) has been studied extensively for the special nonlinearity ϕxt = pxt -γ γ> 0. Indeed, in [7], Edelson proved the existence of a positive solution u ∈ C 2+α loc R d  decaying as x 2-n as x→∞, provided that p ∈ C α loc R d  0 <α< 1px > 0, for x > 0 and satisfying the growth condition  ∞ 0 t n-1+γn-2 max x=t px dt< ∞ (2) In [10], Lazer and Mckenna proved the existence and the uniqueness of a positive solution u ∈ C 2+α loc ∩ C   for the problem (1) provided that  is a bounded domain and ϕxt = pxt -γ γ> 0 with p ∈ C α    and positive on  . In [8], Lair and Shaker proved the result of [10] in R d where 522 0022-247X/01 $35.00 Copyright  2001 by Academic Press All rights of reproduction in any form reserved.