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We consider the variational problem of maximizing the weighted equilibriumGreen's energy of a distribution of charges free to move in a subset of theupper half-plane, under a particular external field. We show that this problemadmits a solution and that, under some conditions, this solution is an S-curve(in the sense of Gonchar-Rakhmanov). The above problem appears in the theory ofthe semiclassical limit of the integrable focusing nonlinear Schr\"odingerequation. In particular, its solution provides a justification of a crucialstep in the asymptotic theory of nonlinear steepest descent for the inversescattering problem of the associated linear non-self-adjoint Zakharov-Shabatoperator and the equivalent Riemann-Hilbert factorization problem.

Existence and regularity for an energy maximization problem in two dimensions