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

Woocheol Choi; Woocheol Choi

doi:10.3934/math.20231332

AIMS Mathematics

2023, Volume 8, Issue 11: 26134-26152. doi: 10.3934/math.20231332

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Theory article

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

Woocheol Choi ^,

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

Received: 10 July 2023 Revised: 01 September 2023 Accepted: 05 September 2023 Published: 12 September 2023
MSC : 35B33, 35J15, 35J60

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.
- blow-up analysis,
- polygonal domains,
- Lane-Emden-Fowler equation
Citation: Woocheol Choi. Energy minimizing solutions to slightly subcritical elliptic problems on nonconvex polygonal domains[J]. AIMS Mathematics, 2023, 8(11): 26134-26152. doi: 10.3934/math.20231332

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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.

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