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

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, 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.