Nonexistence of bounded energy solutions for a fourth order equation on thin annuli (Q1767316)

The authors deal with the problem \[ \begin{gathered} \Delta^2 u_\varepsilon= u_\varepsilon^{{n+4\over n-4}}\quad\text{in }A_\varepsilon,\\ u_\varepsilon> 0\quad\text{in }A_\varepsilon,\\ u_\varepsilon= \Delta u_\varepsilon= 0\quad\text{on }\partial A_\varepsilon,\end{gathered}\tag{1} \] where \(\{A_\varepsilon\subset\mathbb R^n, \varepsilon> 0\}\) is a family of bounded annulus shaped domains such that \(A_\varepsilon\) becomes ``thin'' as \(\varepsilon\to 0\). The main result of the authors state that, if \(u\geq 6\), then there exists \(\varepsilon_0> 0\) such that, for any \(\varepsilon< \varepsilon_0\) the problem (1) has no solution such that \(\int_{A_\varepsilon} |\Delta u_\varepsilon|^2 dx\leq C\), where \(C\) is any constant.