Existence and nonexistence of positive solutions of quasi-linear elliptic equations with gradient terms

TL;DR

This paper investigates conditions for the existence or nonexistence of positive solutions to certain coercive quasi-linear elliptic equations with gradient terms in Euclidean space, highlighting differences based on the parameter p.

Contribution

It provides generalized Keller-Osserman type integral conditions for these equations, accounting for the degeneracy when p varies across 2.

Findings

Derived integral conditions for solution existence

Identified different conditions for p ≥ 2 and p ≤ 2

Extended classical results to quasi-linear cases with gradient terms

Abstract

We study the existence and nonexistence of positive solutions in the whole Euclidean space of coercive quasi-linear elliptic equations such as \[ \Delta_p u = f(u)\pm g(\left|\nabla u\right|) \] where $f\in C([0,\infty))$ and $g\in C^{0,1}([0,\infty)) $ are strictly increasing with $ f(0)=g(0)=0$. Among other things we obtain generalized integral conditions of Keller-Osserman type. In the particular case of plus sign on the right-hand side we obtain that different conditions are needed when $p\geq 2$ or $p\leq 2$, due to the degeneracy of the operator.