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

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 ${\varepsilon }\to 0$.