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Keywords:
elliptic equation; exponential nonlinearity; scalar Chern-Simons equation; signed measure
elliptic equation; exponential nonlinearity; scalar Chern-Simons equation; signed measure
Summary:
We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation \[ -\Delta u + {\rm e}^u({\rm e}^u-1) =\mu \quad \mbox {in}\ \Omega \] 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 $u^{\#}$.
References:
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