Interior regularity of doubly weighted quasi-linear equations

In this article we study the quasi-linear equation $$\mathrm{d}\mathrm{i}\mathrm{v}\mathcal{A}(x,u,\mathrm{\nabla }u)=\mathcal{B}(x,u,\mathrm{\nabla }u)\text{in }\mathrm{\Omega },u\in H_{loc}^{1,p}(\mathrm{\Omega };w_{1}dx)$$ where $\mathcal{A}$ and $\mathcal{B}$ are functions satisfying $\mathcal{A}(x,u,\mathrm{\nabla }u)\sim w_1(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u+\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u)$ and $\mathcal{B}(x,u,\mathrm{\nabla }u)\sim w_2(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u+\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u)$ for $p>1$, a $p$-admissible weight function $w_1$, and another weight function $w_2$ compatible with $w_1$ in a suitable sense. We establish interior regularity results of weak solutions and use those results to obtain point-wise asymptotic estimates at infinity for solutions to $$-\mathrm{d}\mathrm{i}\mathrm{v}(w_1\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)=w_2\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u\text{in }\mathrm{\Omega },u\in D^{1,p,w_1}(\mathrm{\Omega })$$ for a critical exponent $q>p>1$ in the sense of Sobolev.