Abstract

In this paper we consider the nonlinear elliptic problem -Delta u + alpha u = g(vertical bar del u vertical bar) + lambda h(x) in Omega, u = 0 on partial derivative Omega, where Omega is a smooth bounded domain of R-N, alpha >= 0, g is an arbitrary C-1 increasing function and h is an element of C-1((Omega) over bar) is non-negative. We completely analyse the existence and non-existence of (positive) classical solutions in terms of the parameter lambda. We show that there exist solutions for every lambda when alpha = 0 and the integral integral(infinity)(1) 1/g(s) ds = infinity, or when alpha > 0 and the integral integral(infinity)(1) s/g(s) ds = infinity. Conversely, when the respective integrals converge and h is non-trivial on partial derivative Omega, existence depends on the size of lambda. Moreover, non-existence holds for large lambda. Our proofs mainly rely on comparison arguments, and on the construction of suitable supersolutions in annuli. Our results include some cases where the function g is superquadratic and existence still holds without assuming any smallness condition on lambda.