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.