Positive solutions for a class of p-Laplace problems involving quasi-linear and semi-linear terms

The aim of this paper is to discuss the positive solutions of the p-Laplace problem -div(vertical bar del u vertical bar(p-2)del) + g(u)vertical bar del u vertical bar(p) = lambda u(q), where p > 1, q > 1, g: [0, infinity) ->. [0, infinity) is a nonnegative continuous function, lambda is a real number. The sufficient condition to have positive solutions of the above problem is g is an element of L-1 (R+). However, if g is not an element of L-1 (R+), there is no solution which belongs to it. Therefore, our results are optimal. (c) 2006 Elsevier Inc. All rights reserved.