Local uniqueness of the magnetic Ginzburg–Landau equation

In this paper, we consider the magnetic Ginzburg–Landau equation: $$\begin{aligned} \left\{ \begin{aligned}-\Delta _A \psi +\frac{\lambda }{2}(|\psi |^2-1)\psi =0\quad \text {in }{\mathbb {R}}^2,\\ \nabla \times \nabla \times A+\text {Im}({\overline{\psi }}\nabla _{A}\psi )=0\quad \text {in }{\mathbb {R}}^2,\\ |\psi |\rightarrow 1\quad \text {as }|x|\rightarrow +\infty , \end{aligned}\right. \end{aligned}$$where $$\lambda >1$$ is a coupling parameter, $$\nabla _A=\nabla -iA$$ and $$\Delta _A=\nabla _A\cdot \nabla _A$$ are, respectively, the covariant gradient and Laplacian. We prove, by perturbation arguments, that the only possible minimizer of the magnetic Ginzburg–Landau functional with degree 1 is the radial solution for $$\lambda $$ sufficiently close to 1.