  We measure the luminosity function of QSOs in the redshift range 3.5 < z <
5.2 for the absolute magnitude interval -21 < M_{145} < -28. Suitable criteria
are defined to select faint QSOs in the GOODS fields, checking their
effectiveness and completeness in detail. The confirmed sample of faint QSOs is
compared with a brighter one derived from the SDSS. Using a Monte-Carlo
technique we estimate the properties of the luminosity function. Our results
show that models based on pure density evolution show better agreement with
observation than models based on pure luminosity evolution, even if a different
break magnitude with respect to z ~ 2.1 is required at 3.5 < z < 5.2. According
to our modeling a faint-end slope steeper than low-redshift observations is
required to reproduce the data, moreover models with a steep bright-end slope
score a higher probability than models with a bright-end flattening.
Determining the faint-end of the luminosity function at these redshifts
provides important constraints on models of the joint evolution of galaxies and
AGNs.
