We present new measurements of the abundance of galaxies with a given circular velocity in the Local Volume: a region centered on the Milky Way Galaxy and extending to distance 10Mpc. The sample of 750 mostly dwarf galaxies provides a unique opportunity to study the abundance and properties of galaxies down to absolute magnitudes MB= -10, and virial masses Mvir= 1e9Msun. We find that the standard LCDM model gives remarkably accurate estimates for the velocity function of galaxies with circular velocities V>60kms and corresponding virial masses Mvir> 3e10Msun, but it badly fails by over-predicting 5 times the abundance of large dwarfs with velocities V= 30-50kms. The Warm Dark Matter models cannot explain the data either, regardless of mass of the WDM particle. Just as in previous observational studies, we find a shallow asymptotic slope dN/dlog V = V**alpha, alpha =-1 of the velocity function, which is inconsistent with the standard LCDM model that predicts the slope alpha =-3. Though reminiscent to the known overabundance of satellites problem, the overabundance of field galaxies is a much more difficult problem. For the LCDM model to survive, in the 10Mpc radius of the Milky Way there should be 1000 dark galaxies with virial mass Mvir= 1e10Msun, extremely low surface brightness and no detectable HI gas. So far none of this type of galaxies have been discovered.

15 pages, 13 figures. Published in MNRAS: 2015, v454, p1798