A Solution of Problem 2.5 by Brezis on the Planar Ginzburg--Landau Equation

Brezis, Merle, and Riviére [Arch. Rational Mech. Anal. 1994] proved that, if a smooth solution $u:\mathbb{R}^2\to\mathbb{C}$ of the planar entire Ginzburg--Landau equation $$ -Δu = u(1-|u|^2)\quad \text{in}\quad \mathbb{R}^2 $$ satisfies the finite potential energy estimate that $$ \int_{\mathbb{R}^