Interior regularity of some weighted quasi-linear equations

TL;DR

This paper investigates the interior regularity of solutions to certain weighted quasi-linear equations involving p-Laplacian type operators with weights, providing new regularity results and asymptotic estimates for solutions.

Contribution

It establishes interior regularity results for weak solutions of weighted quasi-linear equations and derives point-wise asymptotic estimates for related nonlinear problems.

Findings

Proved interior regularity of weak solutions.

Derived asymptotic estimates for solutions with critical exponents.

Extended regularity theory to weighted quasi-linear equations.

Abstract

In this article we study the quasi-linear equation \[ \left\{ \begin{aligned} \mathrm{div}\, \mathcal A(x,u,\nabla u)&=\mathcal B(x,u,\nabla u)&&\text{in }\Omega,\\ u\in H^{1,p}_{loc}&(\Omega;wdx) \end{aligned} \right. \] 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 \[ \left\{ \begin{aligned} -\mathrm{div}\,(w|\nabla u|^{p-2}\nabla u)&=w|u|^{q-2}u&&\text{in }\Omega,\\ u\in D^{1,p}&(\Omega,wdx) \end{aligned} \right. \] for a critical exponent $q>p$ in the sense of Sobolev.