Bounded Weak Solutions of Degenerate $p$-Poisson 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.