Existence of solutions to a Neumann boundary value problem with exponential nonlinearity

This paper is concerned with the solutions to the following sinh-Poisson equation with Hénon term$\{\begin{array}{cc}?\mathrm{\Delta }u+u={\epsilon }^2|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}({\text{e}}^u?{\text{e}}^{?u}),u>0,\hfill & \text{in}\mathrm{\Omega },\hfill \\ \frac{?u}{?\nu }=0,\hfill & \text{on}?\mathrm{\Omega },\hfill \end{array}$ where $\mathrm{\Omega }?{\mathbb{R}}^2$ is a bounded, smooth domain, $\epsilon >0$, ${\alpha }_1,...,{\alpha }_n\in (0,\infty )?\mathbb{N}$, 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 ${\epsilon }^2|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}({\text{e}}^u?{\text{e}}^{?u})$ 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 $k(x)=|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}$.