Multiple blow-up phenomena for the sinh-Poisson equation

TL;DR

This paper constructs solutions to the sinh-Poisson equation that blow up at the origin with specific positive and negative masses, providing a complete answer to an open problem in the field.

Contribution

It introduces a method to explicitly construct blow-up solutions with prescribed masses for the sinh-Poisson equation, solving an open problem from prior research.

Findings

Solutions blow up at the origin with quantized masses

Positive mass is 4πk(k-1), negative mass is 4πk(k+1)

Addresses an open problem in the analysis of nonlinear PDEs

Abstract

We consider the sinh-Poisson equation $$(P)_\lambda\quad -\Delta u=\la\sinh u\ \hbox{in}\ \Omega,\ u=0\ \hbox{on}\ \partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\rr^2$ and $\lambda$ is a small positive parameter. If $0\in\Omega$ and $\Omega$ is symmetric with respect to the origin, for any integer $k$ if $\la$ is small enough, we construct a family of solutions to $(P)_\la$ which blows-up at the origin whose positive mass is $4\pi k(k-1)$ and negative mass is $4\pi k(k+1).$ It gives a complete answer to an open problem formulated by Jost-Wang-Ye-Zhou in [Calc. Var. PDE (2008) 31: 263-276].