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

TL;DR

This paper proves the existence of sign-changing bubble tower solutions for sinh-Poisson equations on pierced domains, modeling vortex equilibria in hydrodynamic turbulence, with solutions forming asymptotic bubble towers as parameters tend to zero.

Contribution

It establishes the existence of complex sign-changing solutions with bubble tower structures for sinh-Poisson equations on domains with small holes, extending previous results to more intricate solution profiles.

Findings

Existence of sign-changing bubble tower solutions for sinh-Poisson equations.

Construction of solutions with multiple singular bubbles centered at the same point.

Asymptotic behavior of solutions as the parameter approaches zero.

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 $\Omega_\epsilon:=\Omega\setminus \displaystyle \overline{B(\xi,\epsilon)}$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$ and $B(\xi,\epsilon)$ is a ball centered at $\xi\in \Omega$ with radius $\epsilon>0$. Precisely, given a small parameter $\rho>0$ and any integer $m\ge 2$, there exist a radius $\epsilon=\epsilon(\rho)>0$ small enough such that each sinh-Poisson type equation, either in Liouville form or mean field form, has a solution $u_\rho$ with an asymptotic profile as a sign-changing tower of $m$ singular Liouville bubbles centered at the same $\xi$ and with $\epsilon(\rho)\to 0^+$ as $\rho$ approaches to zero.