American Institute of Mathematical Sciences

Advanced
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

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
[1]

Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357
[2]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[3]

Zongming Guo, Long Wei. A perturbed fourth order elliptic equation with negative exponent. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4187-4205. doi: 10.3934/dcdsb.2018132
[4]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[5]

Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007
[6]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[7]

Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.

[8]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033
[9]

M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure & Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233
[10]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[11]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
[12]

Takahiro Hashimoto. Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations. Conference Publications, 2003, 2003 (Special) : 393-402. doi: 10.3934/proc.2003.2003.393
[13]

Craig Cowan. Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter. Communications on Pure & Applied Analysis, 2016, 15 (2) : 519-533. doi: 10.3934/cpaa.2016.15.519
[14]

Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure & Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191
[15]

M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215
[16]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[17]

T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875
[18]

Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143
[19]

Zongming Guo, Long Wei. A fourth order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2493-2508. doi: 10.3934/cpaa.2014.13.2493
[20]

Paolo Caldiroli. Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $R^n$. Communications on Pure & Applied Analysis, 2014, 13 (2) : 811-821. doi: 10.3934/cpaa.2014.13.811

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences