  We investigate the random bipartite optimal matching problem on a flat torus
in two-dimensions, considering general strictly convex power costs of the
distance. We extend the successful ansatz first introduced by Caracciolo et al.
for the quadratic case, involving a linear Poisson equation, to a non-linear
equation of $q$-Poisson type, allowing for a more comprehensive analysis of the
optimal transport cost. Our results establish new asymptotic connections
between the energy of the solution to the PDE and the optimal transport cost,
providing insights on their asymptotic behavior.
