  In this paper, we study a priori estimates for a first-order mean-field
planning problem with a potential. In the theory of mean-field games (MFGs), a
priori estimates play a crucial role to prove the existence of classical
solutions. In particular, uniform bounds for the density of players'
distribution and its inverse are of utmost importance. Here, we investigate a
priori bounds for those quantities for a planning problem with a non-vanishing
potential. The presence of a potential raises non-trivial difficulties, which
we overcome by exploring a displacement-convexity property for the mean-field
planning problem with a potential together with Moser's iteration method. We
show that if the potential satisfies a certain smallness condition, then a
displacement-convexity property holds. This property enables $L^q$ bounds for
the density. In the one-dimensional case, the displacement-convexity property
also gives $L^q$ bounds for the inverse of the density. Finally, using these
$L^q$ estimates and Moser's iteration method, we obtain $L^\infty$ estimates
for the density of the distribution of the players and its inverse.
