Asymptotic Morse theory for the equation $\D v = 2 v_x \wedge v_y$

Abstract

Given a smooth bounded domain $Ø\subseteq {\mathbb{R}}^2$, we consider the equation $\D v = 2 v_x \wedge v_y$ in $Ø$, where $v:Ø\to {\mathbb{R}}^3$. We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An asymptotic study of the corresponding Euler functional is performed, analyzing multiple-bubbling phenomena. This allows us to settle a particular case of a question raised by H. Brezis and J.M. Coron.