标题： Chern-Simons方程的极限解 显示英文标题

标题： Limit solutions of the Chern-Simons equation

摘要： 我们研究标量Chern-Simons方程$-\Delta u + e^u(e^u-1) = \mu$在给定非负有限测度$\mu$时没有解的情况。通过用一系列非负$L^1$函数或对此方程有解的有限测度近似$\mu$，我们证明了Dirichlet问题的解序列收敛到具有最大可能数据$\mu^# \le \mu$的解，并推导出$\mu^#$关于$\mu$的显式公式。 与数据$(\mu, \nu)$对应的 Chern-Simons 系统表现不同，结论取决于测度$\mu$和$\nu$对单点的测度有多大。 显示英文摘要

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