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

Abstract

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 = λ u q , 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 ∈ L 1 ( R + ) . However, if g ∉ L 1 ( R + ) , there is no solution which belongs to it. Therefore, our results are optimal.