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 $Ω_ε:=Ω\setminus \displaystyle \overline{B(ξ,ε)}$, where $Ω$ is a smooth bounded domain in $\mathbb{R}^2$ and $B(ξ,ε)$ is a ball centered at $ξ\in Ω$ with radius $ε>0$. Precisely, given a small parameter $ρ>0$ and any integer $m\ge 2$, 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 $u_ρ$ with an asymptotic profile as a sign-changing tower of $m$ singular Liouville bubbles centered at the same $ξ$ and with $ε(ρ)\to 0^+$ as $ρ$ approaches to zero.