Interior regularity of some weighted quasi-linear equations
Hernán Castro
TL;DR
The paper develops interior regularity theory for weighted quasi-linear equations driven by a weighted $p$-Laplacian in $p$-admissible weights, placing the problem in weighted Sobolev spaces $D^{1,p,w}(\Omega)$. Using Serrin-type structure conditions and weighted Moser iteration, it proves local boundedness, $L^{s,w}$-integrability, a Harnack inequality, and Hölder continuity for weak solutions. Under a global weighted Sobolev inequality, it then derives decay estimates at infinity for solutions of the weighted critical equation $-\operatorname{div}(w|\nabla u|^{p-2}\nabla u)=w|u|^{q-2}u$ with $q=\chi_w p$, including explicit rates $|u(x)|\le C|x|^{-(D_w-p)/p-\lambda}$ and later improvements to $|u(x)|\le C|x|^{-(D_w-p)/(p-1)+\varepsilon}$ for specific weights. For unweighted, monomial, and power-type weights, barrier arguments and comparison principles yield sharper decay, demonstrating how the weight geometry governs asymptotic behavior. The results extend interior regularity and decay analysis to weighted settings and connect to extremals of weighted Sobolev inequalities, with implications for weighted PDEs and related variational problems.
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.