Limit solutions of the Chern–Simons equation

We investigate the scalar Chern–Simons equation in cases where there is no solution for a given nonnegative finite measure datum. Approximating the datum by a sequence of nonnegative integrable functions or finite measures for which this equation has a solution, we show that the sequence of solutions of the Dirichlet problem converges to the solution with largest possible datum. The counterpart for the Chern-Simons system behaves differently and the conclusion depends on how much the measures charge singletons