Aims:In this work the luminosity function of QSOs is measured in the redshift range 3.5<z<5.2 for the absolute magnitude interval -21<M145<-28. Determining the faint-end of the luminosity function at these redshifts provides important constraints on models of the joint evolution of galaxies and AGNs. Methods: We have defined suitable criteria to select faint QSOs in the GOODS fields, checking their effectiveness and completeness in detail. A spectroscopic follow-up of the resulting QSO candidates was carried out. The confirmed sample of faint QSOs is compared with a brighter one derived from the SDSS. We used a Monte-Carlo technique to estimate the properties of the luminosity function, checking various parameterizations for its shape and evolution. Results: Models based on pure density evolution show better agreement with observation than do models based on pure luminosity evolution. However, a different break magnitude with respect to z˜ 2.1 is required at 3.5<z<5.2. Models with a st