Mean field equation for equilibrium vortices with neutral orientation

This paper is concerned with the equilibrium mean field equation of many vortices of the perfect fluid with neutral orientation, −Δv=λ(ev∫Ωevdx−e−v∫Ωe−vdx) in ΩΩ, v=0v=0 on ∂Ω∂Ω, where Ω⊂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.