Suppose $\Omega \subset \mathbb{R}^n$ is a domain and $u$ is a function on it such that $\Delta u$ converges to $0$ at every boundary point of $\Omega$, can we say from here that $u$ extends continuously to the boundary of $\Omega$? Does it follow from some elliptic regularity theory ? Do we need some condition on boundary of $\Omega$ for this to happen ?
Is it possible to get some concrete reference on this (like some particular theorem in some books/notes)?