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

TL;DR

This paper establishes conditions for the existence or nonexistence of positive radial solutions to a quasilinear Dirichlet problem involving the m-Laplacian with diffusion, introducing a new critical exponent that delineates solution regimes.

Contribution

It introduces a new critical exponent for positive radial solutions of a quasilinear Dirichlet problem with diffusion, extending previous results to include diffusion effects.

Findings

Derived a new critical exponent $m^*_{\alpha,\beta,\gamma}$ for solution existence.

Established existence and nonexistence results based on the critical exponent.

Utilized advanced techniques like blow-up methods and Liouville theorems.

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 $\big(a(|x|)+g(u)\big)^{-\gamma}$, with $\gamma>0$ and $a$, $g$ positive functions satisfying natural growth conditions, are proved. Precisely, we obtain a new critical exponent $m^*_{\alpha,\beta,\gamma}$, 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.