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(vertical bar del u vertical bar(m-2)del u) + f(x, u, del u) = 0 in Omega, u = 0 on partial derivative Omega, where Omega subset of R-n (n >= 2) is a connected smooth domain, and the exponent m is an element of (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 not equal 2. (C) 2012 Elsevier Inc. All rights reserved.