An L∞-estimate for solutions to p-Laplacian type equations using an obstacle approach and applications

Abstract

In this paper, we will address a version of the $L^{\infty }$-estimate for weak solutions of $p-$Laplacian type equations of the form $-diva\left(x,\nabla u\right)=f\left(x\right)\text{in}\Omega ,$where $1<p<\infty$, $\Omega \subset R^n$ ($n⩾2$) is a suitable domain (possibly unbounded with finite measure), $f\in L^q\left(\Omega \right)$ with $q>\frac{n}{p}$ and $q\ge \frac{p}{p-1}$ and $a:\Omega \times R^n\to R^n$ is a continuous vector field satisfying certain structural properties. Unlike the non-variational scenario, our approach is based on analyzing an obstacle problem associated with our equation and exploring its intrinsic qualitative properties. Additionally, in some scenarios, our estimates bring to light new geometric quantities that are not present in the results of the classic literature. At the end, we will present some applications of our estimates, which may be essential in certain contexts of quasi-linear PDEs and related topics.