**Background**

In [Berestycki and Lions](http://www.ams.org/mathscinet-getitem?mr=695535) it is asserted that (on page 316), if I am not misreading, that the "ground state", i.e. action minimizer among nontrivial solutions, corresponding to the action 
$$ S[u] = \int_{\mathbb{R}^d} |\nabla u|^2 + G(u) dx $$
where $G$ satisfies certain conditions must be signed and spherically symmetric. In the paper they showed that under certain conditions on $G$ that such a ground state (that is, signed and spherically symmetric) exists; but they do not show that all grounds states must have these symmetries. 

The only reference I can find on this claim is the paper of [Coleman, Glaser, Martin]( http://www.ams.org/mathscinet-getitem?mr=468913); but I am not entirely convinced that they have established the necessity. Their proof uses the fact that a ground state must be a minimizer of $\int |\nabla u|^2$ under the constraint that $\int G(u)$ is fixed. They then use the Polya-Szego principle: under spherical rearrangements the latter integral is unchanged, while the former can only decrease. 

But the decrease is not necessarily strict. [Brothers and Ziemer](http://www.ams.org/mathscinet-getitem?mr=929981) gave a counterexample in the case that the distribution function of $u$ is not absolutely continuous. 

**My question**

I know how to complete the proof and get that ground states must be spherically symmetric, and monotonic radially, *provided one assume that $G$ is $C^{1,1}$*. This one can do by unique continuation principles for elliptic PDEs or equivalently by uniqueness theorems for ODEs. But in Berestycki-Lions or in Coleman-Glaser-Martin, $G$ is only assumed to be $C^1$, for which non-uniqueness, at least in the case of ODEs, is well known as a possibility. 

So, is the "uniqueness" statement true for $G$ merely $C^1$? Are there known counterexamples?