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

We study the boundary value problem of the quasi-linear elliptic equationdiv(|∇u|m−2∇u)+f(x,u,∇u)=0in Ω,u=0on ∂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.