Multi-bubble solutions for the Dirichlet problem of the $H$-system with higher degree

Abstract

We consider a Dirichlet problem of the $H$-system \begin{equation*} \begin{cases} \Delta v = 2v_x\wedge v_y ~& \text{ in }\mathcal{D},\\ v=\varepsilon \tilde g ~& \text{ on }\partial{\mathcal{D}}, \end{cases} \end{equation*} where $\mathcal D\subset \mathbb{R}^2$ is the unit disk, $v:\mathcal D\to \mathbb{R}^3$, and $\tilde g:\partial \mathcal D\to \mathbb{R}^3$ is a given smooth map. As $\varepsilon\to 0^+$, we construct multi-bubble solutions concentrating at distinct points, taking around each point the profile of degree 2 $H$-bubble. This gives a partial answer to a conjecture due to Brezis-Coron and Chanillo-Malchiodi concerning the limiting configuration in the case of higher degrees. This seems to be the first construction in employing higher-degree harmonic maps as the primary configurations.