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

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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 $\varepsilon\to 0$.
    Mathematics Subject Classification: 5J60, 35J65, 58E05.


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