Existence and nonexistence of positive radial solutions of a quasilinear Dirichlet problem with diffusion

Abstract

In this paper existence and nonexistence results of positive radial solutions of a Dirichlet m-Laplacian problem with different weights and a diffusion term inside the divergence of the form a(|x|) + g(u)−γ , with γ > 0 and a, g positive functions satisfying natural growth conditions, are proved. Precisely, we obtain a new critical exponent m * α,β,γ , which extends the one relative to case with no diffusion and it divides existence from nonexistence of positive radial solutions. The results are obtained via several tools such as a suitable modification of the celebrated blow up technique, Liouville type theorems, a fixed point theorem and a Pohožaev-Pucci-Serrin type identity.