On the asymptotic analysis of $H$-systems. II. The construction of large solutions

Abstract

Let $\Omega\subset\mathbb R^2$ be a bounded domain, $\gamma\in C^{3,\alpha}(\partial\Omega;\mathbb R^3)$ ($0 <\alpha <1$) and $H>0$. Let ${h_{\gamma}}$ be the harmonic extension of $\gamma$ in $\Omega$. We show that if $a_0\in\Omega$ is a regular point of ${h_{\gamma}}$ and a nondegenerate critical point of $K(\cdot,\Omega)$ introduced in part I of this paper [3], then for small $H$, there exists a large solution ${\overline{u}_H}$ to the $H$-system $$\Delta u=2Hu_{x_1}\wedge u_{x_2}\quad\text{in $\Omega$}, \qquad u=\gamma\quad \text{on $\partial\Omega$.}$$ Moreover, ${\overline{u}_H}$ blows up (in the sense of part I) at exactly one point $a_0$ as $H\to 0$.