The question concerns the Sobolev continuous injection $H^s \to L^p$ on a bounded $C^2$ domain $\Omega$. See here for the details. Are there any examples of function $u$, showing that the critical value of $p$, given by $$ p_c = \frac{2d}{d-2s}, $$ is optimal? i.e. we cannot have continuous embedding for larger values of $p$.
On $\mathbb R^n$, it is very easy to find an example: just consider the dilation group, and show that we can keep the $H^s$ norm the same whilst pushing $L^p$ norm to $\infty$. However, such a non-compact, unbounded group is not that easily available for a bounded domain.