Limit solutions of the Chern-Simons equation

We investigate the scalar Chern-Simons equation $-\Delta u + e^u(e^u-1) = \mu$incaseswherethereisnosolutionforagivennonnegativefinitemeasure$\mu$\mathrm{.}Approximating$\mu$byasequenceofnonnegative$L^1$ 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 \mu^# \le \mu and we derive an explicit formula of \mu^# in terms of $\mu$. The counterpart for the Chern-Simons system with datum $(\mu ,\nu )$ behaves differently and the conclusion depends on how much the measures $\mu$ and $\nu$ charge singletons.