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Singular limits in Liouville-type equations

Authors:	Manuel del Pino Michal Kowalczyk Monica Musso

Institution:	(1) Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile;(2) Department of Mathematical Sciences, Kent State University, OH 44242 Kent, USA;(3) Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy;(4) Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avda. Vicuña Mackenna 4860, Macul, Chile

Abstract:	We consider the boundary value problem $\Delta u + \varepsilon ^{2} k{\left( x \right)}e^{u} = 0$ in a bounded, smooth domain $\Omega$ in $\mathbb{R}^{{\text{2}}}$ with homogeneous Dirichlet boundary conditions. Here $\varepsilon > 0,k(x)$ is a non-negative, not identically zero function. We find conditions under which there exists a solution $u_{\varepsilon }$ which blows up at exactly m points as $\varepsilon \to 0$ and satisfies $\varepsilon ^{2} {\int_\Omega {ke^{{u_{\varepsilon } }} \to 8m\pi } }%$ . In particular, we find that if $k\in C^2(\bar\Omega)$ , $\inf _{\Omega } k > 0$ and $\Omega$ is not simply connected then such a solution exists for any given $m \ge 1$ Received: 11 February 2004, Accepted: 17 August 2004, Published online: 22 December 2004

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