Separable solutions of quasilinear Lane-Emden equations

TL;DR

This paper studies solutions to quasilinear Lane-Emden equations in conical domains, establishing existence results using degree theory and homotopy, and identifying non-existence in certain critical cases through integral identities.

Contribution

It introduces a method to find separable solutions in cones for quasilinear equations and provides conditions for existence and non-existence.

Findings

Existence of solutions in cones for certain parameter ranges.

Non-existence results in critical cases.

Reduction of the PDE to an elliptic problem on the sphere.

Abstract

For $0 < p-1 < q$ and $\ge=\pm 1$, we prove the existence of solutions of $-\Gd_pu=\ge u^q$ in a cone $C_S$, with vertex 0 and opening $S$, vanishing on $\prt C_S$, under the form $u(x)=|x|^\gb\gw(\frac{x}{|x|})$. The problem reduces to a quasilinear elliptic equation on $S$ and existence is based upon degree theory and homotopy methods. We also obtain a non-existence result in some critical case by an integral type identity.