  We prove uniqueness for the Vlasov-Poisson system in two and three dimensions
under the condition that the $L^p$ norms of the macroscopic density growth at
most linearly with respect to $p$. This allows for solutions with logarithmic
singularities. We provide explicit examples of initial data that fulfill the
uniqueness condition and that exhibit a logarithmic blow-up. In the
gravitational two-dimensional case, such states are intimately related to
radially symmetric steady solutions of the system. Our method relies on the
Lagrangian formulation for the solutions, exploiting the second-order structure
of the corresponding ODE.
