It's obvious that for a standard bounded domain $\Omega$ the interpolation inequality of Gagliardo-Nirenberg for a special case, can be written as the following:
If $D^m v$ and $v$ belong to $L^p(\Omega)$ then $D^r v$ belongs to $L^p(\Omega)$ for $r=0,\cdots,m-1$. In addition, we have: $$ \| D^r v \|_{L^p(\Omega)} \leq C_{\Omega} \left( \| D^{m} v \|_{L^p(\Omega)} + \| v \|_{L^p(\Omega)} \right). $$
Now, if the support of $v$ is included in a compact $K \subset \Omega$, will the constant $C$ depend on $\Omega$ or on the support $K$ of the function $v$?
For compactly supported functions, such a Sobolev embedding constant is universal (it depends only on the parameters, not on the support of the functions).
In the case mentioned above, this follows from the fact that the scaling-invariant Gagliardo-Nirenberg interpolation inequality reads as follows: there exists $C > 0$ such that, for every $v \in C^\infty_c(\mathbb{R}^n;\mathbb{R})$, $$ \| D^r v \|_{L^p(\mathbb{R}^n)} \leq C \| D^m v \|_{L^p(\mathbb{R}^n)}^{\frac r m} \| v \|_{L^p(\mathbb{R}^n)}^{1-\frac r m}. $$ Then, the associated additive Sobolev inequality follows by Young's inequality for products, with a constant which depends neither on $K$ nor on $\Omega$, as long as $v$ is indeed compactly supported inside $\Omega$.
When $v$ is not compactly supported inside $\Omega$, the scaling invariance is lost and the constant may depend on the size of $\Omega$.