On radial solutions for a Fully Non Linear degenerate or singular PDE

TL;DR

This paper investigates the existence of non-trivial radial solutions to a class of fully nonlinear degenerate or singular PDEs in an annular domain, extending the understanding of such equations with nonlinear gradient terms.

Contribution

It establishes the existence of positive and negative radial solutions for a class of fully nonlinear PDEs with degenerate or singular behavior, under specific conditions on parameters.

Findings

Existence of positive radial solutions.

Existence of negative radial solutions.

Solutions are in the sense of Birindelli and Demengel.

Abstract

In this paper we consider equations $-| \nabla u |^\alpha F ( D^2 u) = |u|^{p-1} u $ in an annulus. $F$ is Fully Nonlinear Elliptic, $\alpha$ is some real $> -1$ and $p > 1+ \alpha$. The solutions are intended in the sense of the definition given by Birindelli and Demengel \cite{BD1}. We prove the existence of non trivial positive/negative radial solutions.