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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 $$H\to 0$$.

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