Advances in Differential Equations

On a Yamabe-type problem on a three-dimensional thin annulus

M. Ben Ayed, K. El Mehdi, M. Hammami, and M. Ould Ahmedou

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the problem: $ (P_{\varepsilon}):\, -\Delta u_\varepsilon = u_\varepsilon^{5},\, u_\varepsilon >0 $ in $ A_\varepsilon; \, u_\varepsilon= 0\,\, \mbox{ on } \partial A_\varepsilon $, where $\{A_{\varepsilon } \subset {\mathbb{R}}^3 : {\varepsilon } >0\}$ is a family of bounded annulus-shaped domains such that $A_{\varepsilon }$ becomes ``thin'' as ${\varepsilon }\to 0$. We show that, for any given constant $C>0,$ there exists $\varepsilon_0>0$ such that for any $\varepsilon < \varepsilon_0$, the problem $(P_{\varepsilon })$ has no solution $u_\varepsilon,$ whose energy, $\int_{A_\varepsilon}|\nabla u_\varepsilon |^2,$ is less than C. Such a result extends to dimension three a result previously known in higher dimensions. Although the strategy to prove this result is the same as in higher dimensions, we need a more careful and delicate blow up analysis of asymptotic profiles of solutions $u_{\varepsilon }$ when ${\varepsilon }\to 0$.

Article information

Source
Adv. Differential Equations Volume 10, Number 7 (2005), 813-840.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867832

Mathematical Reviews number (MathSciNet)
MR2152353

Zentralblatt MATH identifier
1161.35380

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J25: Boundary value problems for second-order elliptic equations 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Citation

Ben Ayed, M.; Hammami, M.; El Mehdi, K.; Ould Ahmedou, M. On a Yamabe-type problem on a three-dimensional thin annulus. Adv. Differential Equations 10 (2005), no. 7, 813--840. https://projecteuclid.org/euclid.ade/1355867832.


Export citation