Solvability of nonlinear elliptic equations with gradient terms

TL;DR

This paper investigates conditions for the existence or non-existence of positive solutions to certain nonlinear elliptic equations with gradient terms in the whole Euclidean space, providing sharp criteria especially for power growth cases.

Contribution

It offers new, sharp conditions on functions f and g that determine solvability of nonlinear elliptic equations with gradient terms, improving previous results.

Findings

Complete characterization of solvability for power growth functions

Sharp conditions for existence and non-existence of solutions

Solvability criteria for general coercive equations

Abstract

We study the solvability in the whole Euclidean space of coercive quasi-linear and fully nonlinear elliptic equations modeled on $\Delta u\pm g(|\nabla u|)= f(u)$, $u\ge0$, where $f$ and $g$ are increasing continuous functions. We give conditions on $f$ and $g$ which guarantee the availability or the absence of positive solutions of such equations in $\R^N$. Our results considerably improve the existing ones and are sharp or close to sharp in the model cases. In particular, we completely characterize the solvability of such equations when $f$ and $g$ have power growth at infinity. We also derive a solvability statement for coercive equations in general form.