Nonlinear Analysis 56 (2004) 119–131 www.elsevier.com/locate/na Partial dierential equations involving subcritical, critical and supercritical nonlinearities Sebasti an Lorca a ; ∗; 1 , Pedro Ubilla b; 2 a Departamento de Matem atica Universidad de Tarapac a Casilla 7D, Arica, Chile b Departamento de Matem aticas y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile Received 1 April 2003; accepted 4 September 2003 Abstract We establish the existence of at least one nonnegative solution for the problem (P) -div(a(|∇u|)∇u)= f(u) in ; u =0 on @; where a and f satisfy conditions near zero. Here the novelty is that we do not need restrictions on the nonlinearities at innity. Therefore, we can consider subcritical, critical and supercritical cases. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Positive solution; Quasilinear problems; Critical problems 1. Introduction This article focuses on the existence of solutions for the quasilinear problem (P) -div(a(|∇u|)∇u)= f(u) in ; u =0 on @: Here is a bounded domain in R N , the functions a and f satisfy some hypotheses near zero (cf. hypotheses (a) and (f 1 ) through (f 3 )), and is a large positive parameter. * Corresponding author. E-mail addresses: slorca@uta.cl (S. Lorca), pubilla@lauca.usach.cl (P. Ubilla). 1 Research supported by CNPq, PRONEX and UTA Grants 4731-01 and 4734-02. 2 Research supported by a DICYT-USACH Grant and FONDECYT Grant # 1990183. 0362-546X/$-see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2003.09.002