**Question.** Are all solutions $u: \mathbf{R}^2 \to \mathbf{C}$ of the Ginzburg-Landau equation (1) radially symmetric? What if one imposes additionally that $\int_{\mathbf{R}^2} ( 1 - \lvert u \rvert^2)^2$ < \infty$?

The (non-magnetic) Ginzburg-Landau equation is
\begin{equation}
\tag{1}
- \Delta u = u ( 1 - \lvert u \rvert^2),
\end{equation}
with unknown a complex-valued function $u$.

In 1996, Petru Mironescu showed that the only solutions $u: \mathbf{R}^2 \to \mathbf{C}$ that are *locally minimizing* for the corresponding functional are radially symmetric. My question is whether this is now known to hold for *all critical points*.