= Solution
In the <Landau-Ginzburg theory> functional, $B^2/(2\mu_0)$ is the magnetic field energy density; the covariant-gradient term is the carrier kinetic energy including phase gradients and electromagnetic coupling. The terms $\alpha|\psi|^2+\beta|\psi|^4$ give the local condensation energy. With the normalization in the question, $n=|\psi|^2$ is the carrier density.
For a uniform field-free equilibrium, minimize $\alpha n+\beta n^2$ over $n\geq0$. Stability for large $n$ requires $\beta>0$, and a nonzero minimum requires $\alpha<0$. Therefore
$$
\boxed{n=-\frac\alpha{2\beta}>0.}
$$
The factor two follows from the question's quartic coefficient $\beta$, rather than a convention using $\beta/2$.
Let $D_A=-i\hbar\nabla-qA$. Under the <gauge transformation>,
$$
D_{A+\nabla\Lambda}(e^{iq\Lambda/\hbar}\psi)=e^{iq\Lambda/\hbar}D_A\psi.
$$
Its squared magnitude is unchanged, $|\psi|$ is unchanged, and $B=\nabla\times A$ is unchanged. Thus the full energy has <gauge invariance>.
For uniform $n$, write $\psi=\sqrt n\,e^{i\phi}$. The kinetic density is $n(\hbar\nabla\phi-qA)^2/(2m)$. Varying $A$ with compactly supported variations and integrating the magnetic term by parts gives
$$
\delta E=\int\left[\frac1{\mu_0}\nabla\times B+\frac{nq^2}{m}\left(A-\frac\hbar q\nabla\phi\right)\right]\cdot\delta A\,d^3x.
$$
At a minimum its coefficient vanishes, giving the <London equation>
$$
\boxed{\nabla\times B=-\frac{\mu_0q^2n}{m}\left(A-\frac\hbar q\nabla\phi\right).}
$$
Taking another curl, using $\nabla\cdot B=0$ and a smooth phase so $\nabla\times\nabla\phi=0$, yields
$$
\boxed{\nabla^2B=\frac B{\lambda_L^2},\qquad\lambda_L=\sqrt{\frac{m}{\mu_0q^2n}}.}
$$
For a planar surface this has the decaying solution $B(x)=B(0)e^{-x/\lambda_L}$ inside the material; the growing solution is excluded by boundedness in the bulk. Hence magnetic fields penetrate only a surface layer of thickness $\lambda_L$ and vanish deep in the macroscopic bulk, the <Meissner effect>. This conclusion uses the uniform-density, smooth-phase setting; a vortex has a singular phase and a nonuniform core and is outside those assumptions.
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