We present the result of our low-luminosity quasar survey in the redshift range of 4.5 < z < 5.5 in the COSMOS field. Using the COSMOS photometric catalog, we selected 15 quasar candidates with 22 < i < 24 at z~5, that are ~ 3 mag fainter than the SDSS quasars in the same redshift range. We obtained optical spectra for 14 of the 15 candidates using FOCAS on the Subaru Telescope and did not identify any low-luminosity type-1 quasars at z~5 while a low-luminosity type-2 quasar at z~5.07 was discovered. In order to constrain the faint end of the quasar luminosity function at z~5, we calculated the 1sigma confidence upper limits of the space density of type-1 quasars. As a result, the 1sigma confidence upper limits on the quasar space density are Phi< 1.33*10^{-7} Mpc^{-3} mag^{-1} for -24.52 < M_{1450} < -23.52 and Phi< 2.88*10^{-7} Mpc^{-3} mag^{-1} for -23.52 < M_{1450} < -22.52. The inferred 1sigma confidence upper limits of the space density are then used to provide constrains on the faint-end slope and the break absolute magnitude of the quasar luminosity function at z~5. We find that the quasar space density decreases gradually as a function of redshift at low luminosity (M_{1450} ~ -23), being similar to the trend found for quasars with high luminosity (M_{1450}<-26). This result is consistent with the so-called downsizing evolution of quasars seen at lower redshifts.
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