  In previous works, G. Tomassini and the authors studied and classified
complex surfaces admitting a real-analytic pluri-subharmonic exhaustion
function; let $X$ be such a surface and $D\subseteq X$ a domain admitting a
\emph{continuous} plurisubharmonic exhaustion function: what can be said about
the geometry of $D$? If the exhaustion of $D$ is assumed to be smooth, the
second author already answered this question; however, the continuous case is
more difficult and requires different methods. In the present paper, we address
such question by studying the local maximum sets contained in $D$ and their
interplay with the complex geometric structure of $X$; we conclude that, if $D$
is not a modification of a Stein space, then it shares the same geometric
features of $X$.
