Background

In Berestycki and Lions 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; 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 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?

Background

In Berestycki and Lions 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 (by which I mean that they don't show the "action minimizing solutions" must be both single-signed and spherically symmetric).

The only reference I can find on this claim is the paper of Coleman, Glaser, Martin; 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 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?