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 $-21<M_{145}<-28$. 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 steeper faint-end score a higher probability. We do not find any evidence for a bright-end flattening at redshift $z>3.5$.Conclusions.The estimated space density evolution of QSOs indicates a suppression of the formation and/or feeding of supermassive black holes at these redshifts. The QSO contribution to the UV background is insufficient for ionizing the IGM at $3.5<z<5.2$.