Nonlinear Analysis 66 (2007) 1351–1364 www.elsevier.com/locate/na Subcritical perturbations of a singular quasilinear elliptic equation involving the critical Hardy–Sobolev exponent R.B. Assunc ¸˜ ao a , P.C. Carri˜ ao a , O.H. Miyagaki b,∗ a Departamento de Matem´ atica, Universidade Federal de Minas Gerais 31270-010—Belo Horizonte (MG), Brazil b Departamento de Matem´ atica, Universidade Federal de Vic ¸osa 36571-000—Vic ¸osa (MG), Brazil Received 11 November 2005; accepted 19 January 2006 Abstract In this work we improve some known results for a singular operator and also for a wide class of lower- order terms by proving a multiplicity result. The proof is made by applying the generalized mountain- pass theorem due to Ambrosetti and Rabinowitz. To do this, we show that the minimax levels are in a convenient range by combining a special class of approximating functions, due to Gazzola and Ruf, with the concentrating functions of the best Sobolev constant. c  2006 Elsevier Ltd. All rights reserved. MSC: 35A15; 35B25; 35D33; 35J70 Keywords: Variational methods; Singular perturbations; Critical exponents and degenerate problems 1. Introduction and main results In this work we consider the singular quasilinear elliptic problem  -div[|x | -ap |∇u | p-2 u ]= g(x , u ) +|x | -bq |u | q -2 u in Ω u = 0 on ∂ Ω (1) where Ω ⊂ R N ( N  3) is a bounded domain containing the origin with smooth boundary ∂ Ω , 0  a <( N - p)/ p, a < b < a + 1, d ≡ a + 1 - b, q ≡ Np/[ N - pd ] is the critical ∗ Corresponding author. Tel.: +55 31 389 92 393; fax: +55 31 389 92 393. E-mail addresses: ronaldo@mat.ufmg.br (R.B. Assunc ¸˜ ao), carrion@mat.ufmg.br (P.C. Carri˜ ao), olimpio@ufv.br (O.H. Miyagaki). 0362-546X/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2006.01.027