Energy minimizing solutions to slightly subcritical elliptic problems on nonconvex polygonal domains

In this paper we are concerned with the Lane-Emden-Fowler equation

$$\{\begin{array}{rll}-\mathrm{\Delta }u& =u^{\frac{n+2}{n-2}-\epsilon }& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u& >0& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u& =0& \mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ( $n\ge 3$) is a nonconvex polygonal domain and $\epsilon >0$. We study the asymptotic behavior of minimal energy solutions as $\epsilon >0$ goes to zero. A main part is to show that the solution is uniformly bounded near the boundary with respect to $\epsilon >0$. The moving plane method is difficult to apply for the nonconvex polygonal domain. To get around this difficulty, we derive a contradiction after assuming that the solution blows up near the boundary by using the Pohozaev identity and the Green's function.