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

Abstract

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 $$ \Delta u+h_1e^u-h_2e^{-u}=0 \qquad \mathrm{in}~B_1\subset\mathbb R^2. $$ In particular, we get that the blow-up values are multiple of $8\pi.$ It generalizes the result of Jost, Wang, Ye and Zhou [20] where the extra assumption $h_1 = h_2$ is crucially used.