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)p|∇u|=λuq,−div(|∇u|p−2∇u)+g(u)|∇u|p=λuq, where p>1p>1, q>1q>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+)g∈L1(R+). However, if g∉L1(R+)g∉L1(R+), there is no solution which belongs to it. Therefore, our results are optimal.