A famous radial lemma of Strauss states that
Theorem (Strauss). For $n \geq 2$, every radial function $u \in W^{1,2}(\mathbf R^n)$ is almost everywhere equal to a function $U$, continuous for $x \ne 0$ and such that $$|U(x)| \lesssim |x|^{(1-n)/2}\|u\|_{W^{1,2}(\mathbf R^n)}$$ for $|x| \gg1$.
Berestycki and Lions give a similar result for the homogeneous Sobolev space $\mathcal D^{1,2}_0(\mathbf R^n)$ which is the closure of $C_0^\infty (\mathbf R^n)$ under the norm $$\|u\|_{\mathcal D^{1,2}_0(\mathbf R^n)}=\int_{\mathbf R^n} |\nabla u|^2 dx.$$
Theorem (Berestycki and Lions). For $n \geq 3$, every radial function $u \in \mathcal D^{1,2}_0(\mathbf R^n)$ is almost everywhere equal to a function $U$, continuous for $x \ne 0$ and such that $$|U(x)| \lesssim |x|^{(2-n)/2}\|u\|_{\mathcal D^{1,2}_0(\mathbf R^n)}$$ for $|x| \gg1$.
Since $C_0^\infty (\mathbf R^n)$ is dense in both $W^{1,2}(\mathbf R^n)$ and $\mathcal D^{1,2}_0(\mathbf R^n)$, to prove the above two results, it suffices to consider the case $u \in C_0^\infty (\mathbf R^n)$. What I am interested in is the following: if we use a density argument, we would expect from the Berestycki and Lions lemma the inequality $$|u(x)| \lesssim |x|^{(2-n)/2}\|u\|_{\mathcal D^{1,2}_0(\mathbf R^n)}$$ for $|x| \gg1$, where $u \in \mathcal D^{1,2}_0(\mathbf R^n)$. However, it does not seem to be true because, as far as I know, the constant 1 function belongs to $\mathcal D^{1,2}_0(\mathbf R^n)$ whereas the inequality fails. Hence it is natural to ask when went wrong with the density argument. At least I do not see this is a problem with the Strauss lemma because the constant 1 function does not belong to $W^{1,2}(\mathbf R^n)$.
