Sign-changing bubble tower solutions for sinh-Poisson type equations on pierced domains

Abstract

For asymmetric sinh-Poisson type problems with Dirichlet boundary condition arising as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in hydrodynamic turbulence, we address the existence of sign-changing bubble tower solutions on a pierced domain Ωϵ:=Ω∖B(ξ,ϵ)‾, where Ω is a smooth bounded domain in R2 and B(ξ,ϵ) is a ball centered at ξ∈Ω with radius ϵ>0. Precisely, given a small parameter ρ>0 and any integer m≥2, there exist a radius ϵ=ϵ(ρ)>0 small enough such that each sinh-Poisson type equation, either in Liouville form or mean field form, has a solution uρ with an asymptotic profile as a sign-changing tower of m singular Liouville bubbles centered at the same ξ and with ϵ(ρ)→0+ as ρ approaches to zero.