Global H\"{o}lder solvability of linear and quasilinear Poisson equations

Abstract

We establish an existence result for globally continuous weak solutions to elliptic equations of the $p$-Poisson type. This result significantly improves Theorem 8.30 in Gilbarg-Trudinger (1983) and offers a novel contribution for the classical Poisson equation on Lipschitz domains, ensuring global H\"{o}lder continuity of solutions under a minimal assumption on the right-hand side. Applications of this result to embedding theorems are also discussed.