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 $(a(\mathrm{\mid }x\mathrm{\mid })+g(u){)}^{-\gamma }$, with $\gamma >0$ and $a$, $g$ positive functions satisfying natural growth conditions, are proved. Precisely, we obtain a new critical exponent $m_{\alpha ,\beta ,\gamma }^{\ast }$, 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 Poho\v zaev-Pucci-Serrin type identity.