Mean field equation for equilibrium vortices with neutral orientation Original Research Article

This paper is concerned with the equilibrium mean field equation of many vortices of the perfect fluid with neutral orientation, View the MathML source−Δv=λ(ev∫Ωevdx−e−v∫Ωe−vdx) in ΩΩ, v=0v=0 on ∂Ω∂Ω, where View the MathML sourceΩ⊂R2 is a bounded domain with smooth boundary ∂Ω∂Ω, and λ≥0λ≥0 is a constant. Using the isoperimetric inequality of [T. Suzuki, Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. Inst. H. Poincaré 9 (1992) 367–398] and mean value theorem of [T. Suzuki, Semilinear Elliptic Equations, Gakkotosho, Tokyo, 1994], we prove the linear stability and a priori estimate of any solution under some assumptions on the domain and the parameter λλ, which lead to the uniqueness theorem of the trivial solution on a simply connected domain and the calculation of the Leray–Schauder degree on any domain for λλ in a certain range.