Abstract

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems - ?p u = f? (x, u), u ? W0 1, p (O), where O is a bounded domain in RN, N > p, and ? > 0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti-Brezis-Cerami type in a more general form, namely ? a (x) uq + b (x) ur, where 0 = q < p - 1 < r = p* - 1. Here the coefficient a (x) is assumed to be nonnegative but b (x) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the p-Laplacian context of the Brezis-Nirenberg result on local minimization in W0 1, p and C0 1, a C1, a estimate for equations of the form - ?p u = h (x, u) with h of critical growth, a strong comparison result for the p-Laplacian, and a variational approach to the method of upper-lower solutions for the p-Laplacian. © 2009 Elsevier Inc. All rights reserved.