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We prove existence of Abrikosov vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, with multiple magnetic flux quanta per a fundamental cell. We also revisit the existence proof for the Abrikosov vortex lattices, streamlining some arguments and providing some essential details missing in earlier proofs for a single magnetic flux quantum per a fundamental cell. 1 The Ginzburg-Landau theory is reviewed in every book on superconductivity and most of the books on solid state or condensed matter physics. For reviews of rigorous results see the papers [11,12,20,29] and the books [28,17,21,27] 2 In the problem we consider here it is appropriate to deal with Helmholtz free energy at a fixed average magnetic field b := 1|Ω| Ω curl A, where |Ω| is the area or volume of Ω.

We demonstrate that the reduced Hartree-Fock equation (REHF) with an Anderson type background charge distribution has an unique stationary solution by explicitly computing a screening mass at positive temperature.

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