Existence and nonexistence of solutions for quasilinear elliptic equations

In this paper we study the problem -Delta(p)u = f(x, u, del u) in Omega u = 0 on partial derivative Omega, where Omega subset of R-N is a smooth bounded domain, N greater than or equ al to 2, and Delta(p)u = div(\del u\(p-2) del u) defines the p-Laplacian. W e provide some necessary and sufficient conditions on f under which the pro blem admits a weak solution. For the case p = 2 we obtain more general cond itions on f. The main ingredients are degree theory and the super-subsoluti on method. (C) 2000 Academic Press.