This paper is concerned with the solutions to the following sinh-Poisson equation with Hénon term$\{\begin{array}{cc}-\mathrm{\Delta }u+u={\epsilon }^2|x-q_1{|}^{2{\alpha }_1}\cdots |x-q_n{|}^{2{\alpha }_n}({\text{e}}^u-{\text{e}}^{-u}),u>0,\hfill & \text{in}\mathrm{\Omega },\hfill \\ \frac{\partial u}{\partial \nu }=0,\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^2$ is a bounded, smooth domain, $\epsilon >0$, ${\alpha }_1,...,{\alpha }_n\in (0,\infty )\setminus \mathbb{N}$, and $q_1,...,q_n\in \mathrm{\Omega }$ are fixed. Given any two non-negative integers $k,l$ with $k+l⩾1$, it is shown that, for sufficiently small $\epsilon >0$, there exists a solution $u_{\epsilon }$ for which ${\epsilon }^2|x-q_1{|}^{2{\alpha }_1}\cdots |x-q_n{|}^{2{\alpha }_n}({\text{e}}^u-{\text{e}}^{-u})$ asymptotically (i.e. the limit as $\epsilon \to 0$) develops $k+n$ interior Dirac measures and l boundary Dirac measures. The location of blow-up points is characterized explicitly in terms of Green's function of Neumann problem and the function $k(x)=|x-q_1{|}^{2{\alpha }_1}\cdots |x-q_n{|}^{2{\alpha }_n}$.

这篇论文关注的是以下带有 Hénon 项的 sinh-Poisson 方程的解$\{\begin{array}{cc}-\mathrm{\Delta }你+你={\epsilon }^2|X-q_1{|}^{2{\alpha }_1}\cdots |X-q_n{|}^{2{\alpha }_n}({\text{电子}}^{你}-{\text{电子}}^{-你}),你>0,\hfill & \text{在}\mathrm{\Omega },\hfill \\ \frac{\partial 你}{\partial \nu }=0,\hfill & \text{上}\partial \mathrm{\Omega },\hfill \end{array}$ 在哪里 $\mathrm{\Omega }\subset {\mathbb{电阻}}^2$ 是一个有界的光滑域， $\epsilon >0$, ${\alpha }_1,...,{\alpha }_n\in (0,\infty )\setminus \mathbb{N}$， 和 $q_1,...,q_n\in \mathrm{\Omega }$是固定的。给定任意两个非负整数$克,升$ 和 $克+升⩾1$，这表明，对于足够小的 $\epsilon >0$，有解 ${你}_{\epsilon }$ 为此 ${\epsilon }^2|X-q_1{|}^{2{\alpha }_1}\cdots |X-q_n{|}^{2{\alpha }_n}({\text{电子}}^{你}-{\text{电子}}^{-你})$ 渐近地（即极限为 $\epsilon \to 0$) 发展 $克+n$内部狄拉克测度和l边界狄拉克测度。爆炸点的位置根据诺依曼问题的格林函数和函数进行了明确的表征$克(X)=|X-q_1{|}^{2{\alpha }_1}\cdots |X-q_n{|}^{2{\alpha }_n}$.