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 |∇u|p−2∇u +g(u)|∇u|p = λuq , where p >1, q >1, g : [0,∞)→[0,∞) is a nonnegative continuous function, λ is a real number. The sufficient condition to have positive solutions of the above problem is g ∈ L1(R+). However, if g /∈ L1(R+), there is no solution which belongs to it. Therefore, our results are optimal. © 2006 Elsevier Inc. All rights reserved.