Limit solutions of the Chern-Simons equation

We investigate the scalar Chern-Simons equation $-Δu + e^u(e^u-1) = μ$ in cases where there is no solution for a given nonnegative finite measure $μ$. Approximating $μ$ by a sequence of nonnegative $L^1$ 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 $μ^# \le μ$ and we derive an explicit formula of $μ^#$ in terms of $μ$. The counterpart for the Chern-Simons system with datum $(μ, ν)$ behaves differently and the conclusion depends on how much the measures $μ$ and $ν$ charge singletons.

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