Interior regularity of some weighted quasi-linear equations

Abstract

In this article we study the quasi-linear equation \[ \left\{ \begin{aligned} \mathrm{div}\, \mathcal A(x,u,\nabla u)&=\mathcal B(x,u,\nabla u)&&\text{in }Ω,\\ u\in H^{1,p}_{loc}&(Ω;wdx) \end{aligned} \right. \] where $\mathcal{A}$ and $\mathcal{B}$ are functions satisfying $\mathcal{A}(x,u,\nabla u)\sim \mathcal{B}(x,u,\nabla u)\sim w(|\nabla u{|}^{p-2}\nabla u+|u{|}^{p-2}u)$ for $p>1$ and a $p$-admissible weight function $w$. We establish interior regularity results of weak solutions and use those results to obtain point-wise asymptotic estimates for solutions to \[ \left\{ \begin{aligned} -\mathrm{div}\,(w|\nabla u|^{p-2}\nabla u)&=w|u|^{q-2}u&&\text{in }Ω,\\ u\in D^{1,p}&(Ω,wdx) \end{aligned} \right. \] for a critical exponent $q>p$ in the sense of Sobolev.