On a Yamabe Type Problem on Three Dimensional Thin Annulus

Abstract

We consider a Yamabe type problem on a family $A_{\epsilon }$ of annulus shaped domains of ${\mathbb{R}}^3$ which becomes "thin" as $\epsilon$ goes to zero. We show that, for any given positive constant $C$, there exists ${\epsilon }_0$ such that for any $\epsilon <{\epsilon }_0$, the problem has no solution $u_{\epsilon }$ whose energy 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 $u_{\epsilon }$ when $\epsilon$ goes to zero.

24 pages