A generalized Liouville equation and magnetic stability

Abstract

This work considers two related families of nonlinear and nonlocal problems in the plane ${\mathbb{R}}^2$. The first main result derives the general integrable solution to a generalized Liouville equation using the Wronskian of two coprime complex polynomials. The second main result concerns an application to a generalized Ladyzhenskaya-Gagliardo-Nirenberg interpolation inequality, with a single real parameter ${\hat{I}}^2$ interpreted as the strength of a magnetic self-interaction. The optimal constant of the inequality and the corresponding minimizers of the quotient are studied and it is proved that for ${\hat{I}}^2\ge 2$, for which the constant equals $2\ddot{I}€{\hat{I}}^2$, such minimizers only exist at quantized ${\hat{I}}^2\in 2\mathbb{N}$ corresponding to nonlinear generalizations of Landau levels with densities solving the generalized Liouville equation. This latter problem originates from the study of self-dual vortex solitons in the abelian Chern-Simons-Higgs theory and from the average-field-Pauli effective theory of anyons, i.e. quantum particles with statistics intermediate to bosons and fermions. An immediate application is given to Keller-Lieb-Thirring stability bounds for a gas of such anyons which self-interact magnetically (vector nonlocal repulsion) as well as electrostatically (scalar local/point attraction), thus generalizing the stability theory of the 2D cubic nonlinear Schrödinger equation.

64 pages. V3: some minor corrections and added references