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

2025/04/08 by Fang, Xiang, Wei, Juncheng, Zheng, Youquan +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.05655

Abstract

We consider a Dirichlet problem of the H-system \begincases Δv = 2v_x\wedge v_y ~amp; in D,
v=ε g ~amp; on ∂D, \endcases where \mathcal D⊂ ℝ² is the unit disk, v:\mathcal D→ ℝ³, and g:∂ \mathcal D→ ℝ³ is a given smooth map. As ε→ 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.

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