bounded weak solutions of degenerate p-poisson equations

Sullivan Francis MacDonald, Scott Rodney

Published 2022-10-22Version 1

This paper studies boundedness of weak solutions to $p$-Poisson type equations on a bounded domain $\Omega\subset\mathbb{R}^n$. Given a symmetric, non-negative definite matrix-valued function $Q$ defined on $\Omega$, a weight $v\in L^1_\mathrm{loc}(\Omega)$ comparable to the operator norm of $Q$, and a suitable non-negative function $\tau$, we show that weak solutions $(u,\nabla u)\in QH^{1,p}_0(\Omega,v)$ to the Dirichlet problem \begin{eqnarray} \begin{array}{rccl} -\mathrm{{div}}(\left|\sqrt{Q}\nabla u\right|^{p-2}Q\nabla u)+\tau\left|u\right|^{p-2}uv&=&fv&\textrm{in }\Omega\nonumber u&= & 0&\textrm{on }\partial\Omega \end{array} \end{eqnarray} are \emph{a priori} bounded, provided a Sobolev inequality with gain $\sigma>1$, $$\left(\int_\Omega |\varphi|^{p\sigma}dx\right)^{1/p\sigma} \leq C\left(\int_\Omega |\sqrt{Q}\nabla\varphi|^pdx\right)^{1/p}, $$ holds for every Lipschitz $\varphi$ with compact support in $\Omega$ and the data function $f$ belongs to a weighted Orlicz space $L^\Gamma(\Omega,v)$. The Young function $\Gamma$ is required to satisfy an integral condition that depends on the Sobolev gain factor $\sigma$. In particular, we show that the weak solution satisfies the estimate $$ \|u\|_{L^\infty(\Omega,v)} \leq C\|f\|_{L^\Gamma(\Omega,v)}^\frac{1}{p-1}. $$