We introduce a general \(L_p\) -solvability result for the Poisson equation in non-smooth domains \(\Omega \subset \mathbb {R}^d\) , with the zero Dirichlet boundary condition. Our sole assumption on the domain \(\Omega \) is the Hardy inequality: There exists a constant \(N>0\) such that \(\begin{aligned} \int _{\Omega }\Big |\frac{f(x)}{d(x,\partial \Omega )}\Big |^2\,\textrm{d}x\le N\int _{\Omega }|\nabla f|^2 \,\textrm{d}x\quad \text {for any}\quad f\in C_c^{\infty }(\Omega ). \end{aligned}\) To describe the boundary behavior of solutions in a general framework, we propose a weight system composed of a superharmonic function and the distance function to the boundary. Additionally, we explore applications across a variety of non-smooth domains, including convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, and domains \(\Omega \subset \mathbb {R}^d\) for which the Aikawa dimension of \(\Omega ^c\) is less than \(d-2\) . Using superharmonic functions tailored to the geometric conditions of the domain, we derive weighted \(L_p\) -solvability results for various non-smooth domains and specific weight ranges that differ for each domain condition. Furthermore, we provide an application to the Hölder continuity of solutions.
