Fowler transformation and radial solutions for quasilinear elliptic equations. Part 1: the subcritical and the supercritical case.

We illustrate a method, based on a generalized Fowler transformation, to discuss the existence and the asymptotic behavior of positive radial solutions for the following equation: $$\Delta_p u(\textbf{x})+ f(u,|\textbf{x}|)=0,$$ where $\Delta_p u=div(|Du|^{p-2}Du)$, $\textbf{x} \in \mathbb{R}^n$, $n>p>1$. This approach proves to be particularly useful in the spatial dependent case. Moreover it is a good tool to detect singular and fast decay solutions. We apply it to the case in which $f \ge0$ is either subcritical or supercritical, obtaining structure results for positive solutions and refining the estimates on the asymptotic behavior. The equation has been proposed as a reaction diffusion model for a non-Newtonian fluid and can also be regarded as the constitutive law for a problem in elasticity theory.