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

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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

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

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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.

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