# American Institute of Mathematical Sciences

December  2004, 3(4): 557-580. doi: 10.3934/cpaa.2004.3.557

## Nonexistence of bounded energy solutions for a fourth order equation on thin annuli

 1 Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia, Tunisia 2 Faculté des Sciences et Techniques de Nouakchott, BP 5026, Nouakchott, Mauritania

Received  December 2003 Revised  May 2004 Published  September 2004

In this paper we study the problem $(P_{\varepsilon}): \Delta^2 u_\varepsilon = u_\varepsilon^{\frac{n+4}{n-4}}$, $u_\varepsilon >0$ in $A_\varepsilon$; $u_\varepsilon= \Delta u_\varepsilon= 0$ on $\partial A_\varepsilon$, where $\{A_\varepsilon \subset \mathbb R^n, \varepsilon >0 \}$ is a family of bounded annulus shaped domains such that $A_\varepsilon$ becomes "thin" as $\varepsilon\to 0$. Our main result is the following: Assume $n\geq 6$ and let $C>0$ be a constant. Then there exists $\varepsilon_0>0$ such that for any $\varepsilon <\varepsilon_0$, the problem $(P_{\varepsilon})$ has no solution $u_\varepsilon$, whose energy, $\int_{A_\varepsilon}|\Delta u_\varepsilon |^2$ is less than $C$. Our proof involves a rather delicate analysis of asymptotic profiles of solutions $u_\varepsilon$ when $\varepsilon\to 0$.
Citation: M. Ben Ayed, K. El Mehdi, M. Hammami. Nonexistence of bounded energy solutions for a fourth order equation on thin annuli. Communications on Pure & Applied Analysis, 2004, 3 (4) : 557-580. doi: 10.3934/cpaa.2004.3.557
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