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The non-uniqueness of the limit solutions of the scalar Chern-Simons equations with signed measures

Adilson Eduardo Presoto — 2021

Mathematica Bohemica

We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation $- Δ u + e^{u} (e^{u} - 1) = μ in Ω$ with the Dirichlet boundary condition. Approximating $μ$ by a sequence ${(μ_{n})}_{n \in ℕ}$ of $L^{1}$ functions or finite signed measures such that this equation has a solution $u_{n}$ for each $n \in ℕ$ , we are interested in establishing the convergence of the sequence ${(u_{n})}_{n \in ℕ}$ to a function $u^{#}$ and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by $u^{#}$ .

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