Interior regularity of doubly weighted quasi-linear equations

Abstract

In this article we study the quasi-linear equation \[\mathrm{div}\, \mathcal A(x,u,\nabla u)=\mathcal B(x,u,\nabla u)\quad \text{in }\Omega,\qquad u\in H^{1,p}_{loc}(\Omega;w_1dx)\] where $\mathcal A$ and $\mathcal B$ are functions satisfying $\mathcal A(x,u,\nabla u)\sim w_1(|\nabla u|^{p-2}\nabla u+|u|^{p-2}u)$ and $\mathcal B(x,u,\nabla u)\sim w_2(|\nabla u|^{p-2}\nabla u+|u|^{p-2}u)$ 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 \[-\mathrm{div}\,(w_1|\nabla u|^{p-2}\nabla u)=w_2|u|^{q-2}u\quad \text{in }\Omega,\qquad u\in D^{1,p,w_1}(\Omega)\] for a critical exponent $q>p>1$ in the sense of Sobolev.