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

Abstract

In this paper we are concerned with the Lane-Emden-Fowler equation $ \begin{equation*} \left\{\begin{array}{rll}-\Delta u & = u^{\frac{n+2}{n-2}- \varepsilon}& {\rm{in}}\; \Omega, \\ u&>0& {\rm{in}}\; \Omega, \\ u& = 0& {\rm{on}}\; \partial \Omega, \end{array} \right. \end{equation*} $ where $ \Omega \subset \mathbb{R}^n $ ($ n \geq 3 $) is a nonconvex polygonal domain and $ \varepsilon > 0 $. We study the asymptotic behavior of minimal energy solutions as $ \varepsilon > 0 $ goes to zero. A main part is to show that the solution is uniformly bounded near the boundary with respect to $ \varepsilon > 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.

Keywords

• blow-up analysis
• polygonal domains
• lane-emden-fowler equation