Local uniqueness of the magnetic Ginzburg–Landau equation

Abstract

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.

Appendix

In this appendix, we shall prove Proposition 2.1.

Proof of Proposition 2.1

Let $(\varphi ,B)$ be a solution of (1.1) with $\lambda =1$ such that its degree is N, then under Taubes’s notations and results in [19] (see also [16]), $\varphi =e^{\frac{1}{2}(u+i\theta )}$ with $\theta =\sum _{j=1}^m\arg (x-a_j)$ and $B=\frac{1}{2}(\partial _1\theta +\partial _2u,\partial _2\theta -\partial _1u)$, where $(a_1, a_2, \ldots , a_m)$ is the m vortices of $(\varphi , B)$. Moreover, u also satisfies the following elliptic equation:

$$\begin{aligned} -\Delta u+(e^{u}-1)=-4\pi \sum _{j=1}^m\delta _{a_j}\quad \text {in }{\mathbb {R}}^2, \end{aligned}$$

(3.1)

where $\delta _{a_j}$ is the Dirac function at $a_j$. Now, using this information in the energy functional ${\mathcal {E}}_1(\psi ,A)$, we have

$$\begin{aligned} {\mathcal {E}}_1(\psi ,B)=\int _{{\mathbb {R}}^2}\frac{1}{4}|\nabla u|^2e^u+\frac{1}{8}|\Delta u|^2+\frac{1}{8}(e^u-1)^2. \end{aligned}$$

(3.2)

By Levi’s monotone convergence theorem and (3.1),

$$\begin{aligned} \int _{{\mathbb {R}}^2}|\Delta u|^2=\lim _{\varepsilon \rightarrow 0}\int _{{\mathbb {R}}^2\backslash \cup _{j=1}^mB_\varepsilon (a_j)}|\Delta u|^2=\int _{{\mathbb {R}}^2}(e^u-1)^2. \end{aligned}$$

(3.3)

On the other hand, since $|\varphi |\le 1$ (cf. [20]), by the diamagnetic inequality (cf. [8, (2.3)]), we know that $|\varphi |^2-1=e^{u}-1$ in $H^1({\mathbb {R}}^2)$. Moreover, since $u=-\sum _{j=1}^{m}\ln (1+\frac{\sigma }{|x-a_j|^2})+v$ for some $\sigma >4m$ and smooth v which exponentially decays to zero as $|x|\rightarrow +\infty$ (cf. [19]), $|\nabla u|\sim \frac{1}{|x-a_j|}$ near each vortex point $a_j$ and $|\nabla u|\sim \frac{1}{|x|^3}$ as $|x|\rightarrow +\infty$. Thus, we can multiply (3.1) in ${\mathbb {R}}^2\backslash \cup _{j=1}^mB_\varepsilon (a_j)$ with $e^{u}-1$ and integrate by parts, which implies that

$$\begin{aligned} \int _{{\mathbb {R}}^2\backslash \cup _{j=1}^mB_\varepsilon (a_j)}|\nabla u|^2e^{u}+\int _{{\mathbb {R}}^2\backslash \cup _{j=1}^mB_\varepsilon (a_j)}(e^{u}-1)^2=4m\pi . \end{aligned}$$

Let $\varepsilon \rightarrow 0$ and applying Levi’s monotone convergence theorem yield that

$$\begin{aligned} \int _{{\mathbb {R}}^2}|\nabla u|^2e^{u}+\int _{{\mathbb {R}}^2}(e^{u}-1)^2=4m\pi . \end{aligned}$$

(3.4)

Inserting (3.3) and (3.4) into (3.2) and recalling that ${\mathcal {E}}_1(\psi ,B)=N\pi$ since $(\varphi ,B)$’s degree is N, we must have $m=N$. $\square$