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

M. Ben Ayed; K. El Mehdi; M. Hammami

doi:10.3934/cpaa.2004.3.557

Article Contents

2004, Volume 3, Issue 4: 557-580. Doi: 10.3934/cpaa.2004.3.557

This issue Previous Article Stability of Suliciu model for phase transitions Next Article Ornstein-Zernike behavior for the Bernoulli bond percolation on $\mathbb Z^d$ in the supercritical regime

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
2.
Faculté des Sciences et Techniques de Nouakchott, BP 5026, Nouakchott

Received: November 30, 2003

Revised: April 30, 2004

Published: November 2004

Abstract Related Papers Cited by

Abstract

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:

Article Metrics

HTML views() PDF downloads(60) Cited by(0)

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

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

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

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content