On the asymptotic analysis of \(H\)-systems. I: Asymptotic behavior of large solutions. (Q5954483)

The author considers following so-called an \(H\)-system: NEWLINE\[NEWLINE\begin{gathered} \Delta u= 2Hu_{x_1}\wedge u_{x_2}\quad\text{in }\Omega,\\ u= \gamma\quad\text{on }\partial\Omega,\end{gathered}\tag{1}NEWLINE\]NEWLINE where \(\Omega\subset\mathbb{R}^2\) is a bounded domain and \(\gamma\in C^{3,\alpha}(\partial\Omega, \mathbb{R}^3)\), \(0< \alpha< 1\), \(\Lambda\) is the exterior product in \(\mathbb{R}^3\) and \(H> 0\). The main goal of the author is to find the asymptotic behaviour of large solutions of (1) as \(H\to 0\) aor as \(\gamma\to 0\). The author shows that large solutions blow-up at exactly one point in \(\Omega\). The exact blow-up rate and location of blow-up point are studied as well.