Nature of minimizers for 0 < β < 2

Determine whether, for 0 < β < 2, any minimizer u ∈ H^1(ℝ^2) of γ*(β) := inf{ E_β[u] / ∫|u|^4 : ∫|u|^2 = 1 }, where E_β[u] = ∫ |(∇ + i β A[|u|^2])u|^2 with A[|u|^2](x) = ∫ (x−y)^{⊥}/|x−y|^2 |u(y)|^2 dy, must (i) be non-real-valued and (ii) have zeros (vortices).

Background

The paper introduces the LGN-type magnetic interpolation problem with a self-generated field and defines the optimal constant γ(β) via a variational principle over H^1(ℝ^2). For β ≥ 2, the authors prove that γ(β) = 2πβ and classify all minimizers explicitly; they also show that at β = 2 there exist real minimizers whereas for β > 2 there are no real minimizers and minimizers carry nontrivial vorticity.

In contrast, for 0 < β < 2 they establish refined bounds and partial existence (for sufficiently small β), but the qualitative structure of minimizers in this regime remains unsettled. The authors explicitly pose as an open problem whether minimizers in this range are necessarily non-real and whether they possess zeros (vortices).