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On the asymptotic analysis of \(H\)-systems. II: The construction of large solutions. (English) Zbl 1004.35050
Summary: 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 [T. Isobe, On the asymptotic analysis of \(H\)-systems, I: Asymptotic behavior of large solutions, Adv. Differ. Equ. 6, No. 5, 513-546 (2001; Zbl 1142.35345], then for small \(H\), there exists a large solution \(\overline u_H\) to the \(H\)-system \[ \Delta u=2Hu_{x_1}\wedge u_{x_2}\text{ in } \Omega,\quad u=\gamma \text{ on }\partial\Omega. \] Moreover, \(\overline u_H\) blows up (in the sense of part I) at exactly one point \(a_0\) as \(H\to 0\).

35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35J50 Variational methods for elliptic systems