Multiple blow-up phenomena for the sinh-Poisson equation

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