Mean field equation for equilibrium vortices with neutral orientation

Abstract

This paper is concerned with the equilibrium mean field equation of many vortices of the perfect fluid with neutral orientation, − Δ v = λ ( e v ∫ Ω e v d x − e − v ∫ Ω e − v d x ) in Ω , v = 0 on ∂ Ω , where Ω ⊂ R 2 is a bounded domain with smooth boundary ∂ Ω , and λ ≥ 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 λ , 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 λ in a certain range.