Calculus of Variations and Partial Differential Equations

, Volume 24, Issue 1, pp 47-81

First online:

Singular limits in Liouville-type equations

  • Manuel del PinoAffiliated withDepartamento de Ingeniería Matemática and CMM, Universidad de Chile
  • , Michal KowalczykAffiliated withDepartment of Mathematical Sciences, Kent State University
  • , Monica MussoAffiliated withDipartimento di Matematica, Politecnico di TorinoDepartamento de Matemática, Pontificia Universidad Catolica de Chile

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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 \(m \ge 1\)