Regularity of Singular Solutions to $p$-Poisson Equations

This work showcases level set estimates for weak solutions to the $p$-Poisson equation on a bounded domain, which we use to establish Lebesgue space inclusions for weak solutions. In particular we show that if $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain and $u$ is a weak solution to the Dirichlet problem for Poisson’s equation $$-\mathrm{\Delta }u=f\text{ in }\mathrm{\Omega }$$ $$u=0\text{ on }\mathrm{\partial }\mathrm{\Omega }$$ for $f\in L^q(\mathrm{\Omega })$ with $q<\frac{n}{2}$, then $u\in L^r(\mathrm{\Omega })$ for every $r<\frac{qn}{n-2q}$ and indeed $\mathrm{\parallel }u{\mathrm{\parallel }}_r\le C\mathrm{\parallel }f{\mathrm{\parallel }}_q$. This result is shown to be sharp, and similar regularity is established for solutions to the $p$-Poisson equation including in the edge case $q=\frac{n}{p}$.