Limit solutions of the Chern–Simons equation

Given a bounded domain ΩΩ in R2R2, we investigate the scalar Chern–Simons equation −Δu+eu(eu−1)=μin Ω, in cases where there is no solution for a given nonnegative finite measure μμ. Approximating μμ by a sequence (μn)n∈N(μn)n∈N of nonnegative L1L1 functions or finite measures for which this equation has a solution, we show that the sequence of solutions (un)n∈N(un)n∈N of the Dirichlet problem converges to the solution with largest possible datum μ#≤μμ#≤μ and we derive an explicit formula of μ#μ# in terms of μμ. The counterpart for the Chern-Simons system with datum (μ,ν)(μ,ν) behaves differently and the conclusion depends on how much the measures μμ and νν charge singletons.