Comparison Principle, A.B.P.-type estimates for solutions of quasi-linear elliptic equations in non-divergence form and some implications

TL;DR

This paper develops global gradient estimates for solutions of certain quasi-linear elliptic equations in non-divergence form, including degeneracy and Hamiltonian terms, with applications to non-degeneracy and other properties.

Contribution

It introduces new gradient and non-degeneracy estimates for quasi-linear elliptic equations with degeneracy and Hamiltonian terms.

Findings

Established global gradient estimates for solutions.

Provided non-degeneracy estimates.

Presented applications of the estimates.

Abstract

In this work, we establish global gradient estimates to solutions of quasilinear elliptic models in non-divergence form with general degeneracy law and a Hamiltonian term, given by $$ -\Psi(x, |\nabla u|)\Delta_p^{\mathrm{N}}u(x)+\mathscr{H}(x,\nabla u)=f(x) \quad \mathrm{in} \quad \Omega, \quad \mathrm{for} \,\,\,1<p< \infty, $$ under suitable assumptions on the data of the problem. Particularly, our results are relevant for a class of quasi-linear models with Hamiltonian terms. Additionally, we address non-degeneracy estimates for such solutions and present a couple of applications.