  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.
