ESO deep observations of the optical afterglow of GRB 990510

We present the results of optical observations of the GRB 990510 field carr ied out at different epochs from European Southern Observatory (ESO) telesc opes. Deep observations, down to a limiting magnitude of about 27 and 24 in the Bessel-R and Gunn-I band, respectively, were obtained between May 16 a nd 18 from the ESO NTT-SUSI2 telescope and on May 20 from the ESO 3.6 m (EF OSC2) telescope. These observations, together with other published photomet ric data, allowed to monitor the faint tail of the decaying Optical Transie nt (OT) associated to the GRB 990510. We discuss the light curves in the di fferent filters (V, R and I) in the light of the recently proposed decay la ws. No obvious host associated to the GRB 990510 optical afterglow was foun d in the R and I band images. By comparing the light curves with respect to the theoretical colors of galaxies with different morphology we derived a lower limit of R similar to 26.6 for the host galaxy.