I'm currently working on a project in which I have to establish some estimates for some global Sobolev and Lebesgue norms.
We know that if we have a bounded domain $\Omega$, then for any $q \leq p^*$ there is a $C > 0$ such that:
$$\|u\|_{L^q} \leq C\|\nabla{u}\|_{L^p}\tag{1}$$
for any $u$.
If the domain is not bounded, we have this estimate only for $q=p^*$ (due to Gagliardo-Niremberg Inequality), because we cannot use Holder inequality to bound $L^q$ norms by other such norms with bigger $q$.
However, if we replace $\|\nabla{u}\|_{L^p}$ by $\|u\|_{W^{1,p}}$, the $\|u\|_{L^p}$ appearing on the RHS allows us to establish:
$$\|u\|_{L^q} \leq C(q)\|u\|_{W^{1,p}} \tag{2}$$
for all $p \leq q \leq p^*$.
My question is: do we have such an estimate for unbounded domains, but also for $q < p$? More generally, for which values of $q$ can we establish estimates of the form (1) and (2), when the domain is unbounded? (I'm basically interested in the case $\Omega = \mathbb{R}^3$.)
One word: scaling.
Consider $u_\lambda(x) = u(\lambda x)$ where $\lambda>0$. The Lebesgue norm scales as $$\|u_\lambda\|_p = \lambda^{-n/p}\|u\|_p$$ while the norm of the gradient scales as $$\|\nabla u_\lambda\|_p = \lambda^{1-n/p}\|\nabla u\|_p$$ This immediately shows that $\|\nabla u_\lambda\|_p$ can control $\|u\|_q$ only when $\frac{n}{q} = 1-\frac{n}{p}$, which corresponds to $q=p^*$, the critical exponent.
The full norm of $W^{1,p}$ has a term that scales as $\lambda^{-n/p}$ and another that scales as $\lambda^{1-n/p}$. The sum of such things dominates the powers of $\lambda$ that are between $-n/p$ and $1-n/p$, and none others. This leads to the necessary condition $$-\frac{n}{p}\le -\frac{n}{q} \le 1-\frac{n}{p} $$ that is, $p\le q\le p^* $.