On positive solutions of quasi-linear elliptic equations involving critical Sobolev growth

We study the boundary value problem of the quasi-linear elliptic equation div |∇u|m−2∇u +f (x,u,∇u) = 0 inΩ, u = 0 on∂Ω, where Ω ⊂ Rn (n 2) is a connected smooth domain, and the exponent m ∈ (1,n) is a positive number. Under appropriate conditions on the function f , a variety of results on existence and non-existence of positive solutions have been established. This paper is a continuation of an earlier work Zou (2008) [18] of the author and, in particular, extends earlier results of Brezis and Nirenberg (1983) [3] for the semi-linear case of m = 2, and of Pucci and Serrin (1986) [12] for the quasi-linear case of m = 2. © 2012 Elsevier Inc. All rights reserved.