Classification of blow-up limits for the sinh-Gordon equation

The aim of this paper is to use a selection process and a careful study of the interaction of bubbling solutions to show a classification result for the blow-up values of the elliptic sinh-Gordon equation $$\mathrm{\Delta }u+h_1{e}^u-h_2{e}^{-u}=0\mathrm{i}\mathrm{n}{B}_1\subset {\mathbb{R}}^2\mathrm{.}$$ In particular we get that the blow-up values are multiple of $8\pi \mathrm{.}$ It generalizes the result of Jost, Wang, Ye and Zhou \cite{jwyz} where the extra assumption $h_1 = h_2$ is crucially used.