We construct singular solutions to a semilinear elliptic equation with exponential nonlinearity on $\Omega \subset \mathbb{R}^2$ by a shrinking hole argument, which we call Born–Infeld approximation scheme. With some natural assumptions on the nonlinearity $f (e^u)$, we classify all these singular solutions by the order of $u$ near each singularity. To show the existence of singular solutions, we introduce an inverse problem and utilize a Born–Infeld approximation scheme. Ruling out a possible occurrence of bubbling phenomenon, we show that as the Born–Infeld parameter tends to infinity, solutions of the inverse problem on a subdomain of $\Omega$ with finitely many holes can be used to approximate singular solutions to the original equation, provided that the size of each hole is carefully given and satisfies some constraint condition on the total flux. Our work rigorously justifies the Born–Infeld–Higgs approximation to the Abelian Maxwell–Higgs theory. When $b=1$, we also find finite-energy solutions to the equation of Born–Infeld gauged harmonic maps, which have finitely many magnetic singularities in $\Omega$.

中文翻译：

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一些半线性椭圆方程的奇异解：Born-Infeld逼近的一种方法

我们通过收缩孔参数在$ \ Omega \ subset \ mathbb {R} ^ 2 $上构造具有指数非线性的半线性椭圆方程的奇异解，我们称其为Born-Infeld近似方案。通过对非线性$ f（e ^ u）$的一些自然假设，我们将所有这些奇异解按每个奇异点附近的$ u $的顺序进行分类。为了说明奇异解的存在，我们引入了一个反问题，并利用了Born-Infeld近似方案。排除冒泡现象的可能发生，我们证明，随着Born-Infeld参数趋于无穷大，在$ \ Omega $子域上具有有限多个孔的逆问题的解可用于近似原始方程的奇异解，只要仔细地确定每个孔的大小并满足总通量的某些约束条件即可。我们的工作严格证明了Born-Infeld-Higgs近似于Abelian Maxwell-Higgs理论。当$ b = 1 $时，我们还找到了Born-Infeld规范谐波图方程的有限能量解，该方程在$ \ Omega $中具有有限的磁奇点。