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

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 $$-\mathrm{\Psi }(x,\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }){\mathrm{\Delta }}_p^{\mathrm{N}}u(x)+\mathcal{H}(x,\mathrm{\nabla }u)=f(x)\mathrm{i}\mathrm{n}\mathrm{\Omega },\mathrm{f}\mathrm{o}\mathrm{r}1<p<\mathrm{\infty },$$ 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.