We use molecular dynamics simulations to study polyelectrolytes end-grafted to a surface. Our model includes counterions explicitly and treats the full long-range Coulomb interaction. Strongly charged polyelectrolytes with no added salt are investigated, where the counterions are localized inside the brush and electroneutrality is satisfied locally. At this stage, we do not rule out that finite system size influences the prefactors of our scaling results. Irrespective of this question we find that the brushes are collapsed with a thickness linearly proportional to the chain length and the grafting density. The counterion distribution is strongly inhomogeneous, and counterion condensation can be observed although the Bjerrum length is smaller than the average bond length. The osmotic counterion pressure is much larger than the ideal gas value. Counterion diffusion is anisotropic and is enhanced at higher grafting densities. We compare our results to experiments and to scaling predictions.