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

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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

M. Ben Ayed ^1, , K. El Mehdi ^2, and M. Hammami ^1,

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

Abstract
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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 $\varepsilon\to 0$.

Keywords: Fourth order elliptic equations, critical Sobolev exponent.

Mathematics Subject Classification: 5J60, 35J65, 58E05.

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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