On a Yamabe Type Problem on Three Dimensional Thin Annulus

We consider a Yamabe type problem on a family $A_\epsilon$ of annulus shaped domains of $\R^3$ 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.