Bounded Weak Solutions of Degenerate $p$-Poisson Equations

TL;DR

This paper establishes conditions under which weak solutions to degenerate p-Poisson equations are bounded and exponentially integrable, using an iterative De Giorgi method, with implications for solutions with data in Orlicz spaces.

Contribution

It provides new sufficient conditions for boundedness and exponential integrability of solutions to degenerate p-Poisson equations, extending previous results to more general settings.

Findings

Weak solutions are bounded under certain conditions.

Solutions exhibit exponential integrability when data is in specific Orlicz spaces.

The iterative De Giorgi method is effectively applied to degenerate equations.

Abstract

In this work we study global boundedness and exponential integrability of weak solutions to degenerate $p$-Poisson equations using an iterative method of De Giorgi type. Given a symmetric, non-negative definite matrix valued function $Q$ defined on a bounded domain $\Omega\Subset\mathbb{R}^n$, a weight function $v\in L^1_\textrm{loc}(\Omega,dx)$, and a suitable non-negative function $\tau$, we give sufficient conditions for any weak solution to the Dirichlet problem \begin{align*} \begin{array}{rccl} -\displaystyle\frac{1}{v}\mathrm{{div}}\left(\left|\sqrt{Q}\nabla u\right|^{p-2}Q\nabla u\right)+\tau\left|u\right|^{p-2}u&=&f&\textrm{in }\Omega, \end{array} \end{align*} \begin{align*} \begin{array}{rccl} u&= & 0&\textrm{on }\partial\Omega \end{array} \end{align*} to be bounded and exponentially integrable when the data function $f$ belongs to an appropriate Orlicz space.