  The intergalactic medium (IGM) at $z\sim$ 5 to 6 is largely ionized, and yet
the main source for the IGM ionization in the early universe is uncertain. Of
the possible contributors are faint quasars with $-26 \lesssim M_{\rm 1450}
\lesssim -23$, but their number density is poorly constrained at $z\sim5$. In
this paper, we present our survey of faint quasars at $z\sim5$ in the European
Large-Area {\it ISO} Survey-North 1 (ELAIS-N1) field over a survey area of 6.51
deg$^2$ and examine if such quasars can be the dominant source of the IGM
ionization. We use the deep optical/near-infrared data of the ELAIS-N1 field as
well as the additional medium-band observations to find $z \sim 5$ quasars
through a two-step approach using the broadband color selection, and SED
fitting with the medium-band information included. Adopting Bayesian
information criterion, we identify ten promising quasar candidates. Spectra of
three of the candidates are obtained, confirming all of them to be quasars at
$z\sim5$ and supporting the reliability of the quasar selection. Using the
promising candidates, we derive the $z\sim5$ quasar luminosity function at $-26
\lesssim M_{\rm 1450} \lesssim -23$. The number density of faint $z\sim5$
quasars in the ELAIS-N1 field is consistent with several previous results that
quasars are not the main contributors to the IGM-ionizing photons at $z\sim5$
