Nonlinear Analysis 53 (2003) 147–155 www.elsevier.com/locate/na Remarks on the existence of global minimizers for the Ginzburg–Landau energy functional T. Giorgi ∗ , R.G. Smits Mathematics Department, Towson University, 8000 York Road, Towson, MD 21252-0001, USA Received 28 July 2000; received in revised form 20 February 2001; accepted 21 May 2001 Keywords: Superconductivity; Ginzburg–Landau energy; Existence global minimizers; Weighted Sobolev spaces 1. Introduction We present a proof of existence in a weighted Sobolev space of global minimizers for the Ginzburg–Landau energy functional of superconductivity in R 2 . The use of weighted Poincar e inequalities allows for a unied approach with the case of the functional dened on R 3 . The Ginzburg–Landau free energy functional for a low temperature superconductor which occupies all of R n (n =2; 3) can be written in non-dimensional form as follows (see [3] for a mathematical review of the subject): G(; A; H a ):=  R n  1 2 (1 -| | 2 ) 2 +      i  ∇ + A     2 + |curl A - H a | 2  dx: (1.1) In our notation,  is the complex valued superconducting order parameter, A is the magnetic potential, H a represents the external applied magnetic eld and  is the so-called Ginzburg–Landau parameter. The unknowns of the problem are  and A, where | | is proportional to the density of the superconducting electrons pairs, curl A in R 3 is the induced magnetic eld, and j:=-(i= 2)( ∗ ∇ - ∇ ∗ ) -A| | 2 in R n is the supercurrent density. Here ∗ represents complex conjugation. * Corresponding author. Tel.: +1-410-830-3091; fax: +1-410-830-4149. E-mail addresses: tgiorgi@towson.edu (T. Giorgi), rsmits@towson.edu (R.G. Smits). 0362-546X/03/$-see front matter c  2003 Elsevier Science Ltd. All rights reserved. PII:S0362-546X(01)00800-8