Existence and non-existence of minimizers for Poincar\'e-Sobolev inequalities

In this paper we study the existence and non-existence of minimizers for a type of (critical) Poincar'{e}-Sobolev inequalities. We show that minimizers do exist for smooth domains in ${\mathbb{R}}^d$, an also for some polyhedral domains. On the other hand, we prove the non-existence of minimizers in the rectangular isosceles triangle in ${\mathbb{R}}^2$.