Mathematica Bohemica, Vol. 146, No. 3, pp. 235-249, 2021
The non-uniqueness of the limit solutions of the scalar Chern-Simons equations with signed measures
Adilson Eduardo Presoto
Received December 14, 2018. Published online September 25, 2020.
Abstract: We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation
$-\Delta u + {\rm e}^u({\rm e}^u-1) =\mu\quad in \Omega$ with the Dirichlet boundary condition. Approximating $\mu$ by a sequence $(\mu_n)_{n \in\mathbb N}$ of $L^1$ functions or finite signed measures such that this equation has a solution $u_n$ for each $n\in\mathbb{N}$, we are interested in establishing the convergence of the sequence $(u_n)_{n\in\mathbb{N}}$ 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 $u^#$.
Keywords: elliptic equation; exponential nonlinearity; scalar Chern-Simons equation; signed measure
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Affiliations: Adilson Eduardo Presoto, Federal University of São Carlos, Rodovia Washington Luis, Km 235, 13565-905 São Carlos-SP, Brazil, e-mail: presoto@dm.ufscar.br
