Singular limits in Liouville-type equations

  • Manuel del Pino
  • Michal Kowalczyk
  • Monica Musso
Article

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 \(m \ge 1\)

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Copyright information

© Springer-Verlag Berlin/Heidelberg 2005

Authors and Affiliations

  • Manuel del Pino
    • 1
  • Michal Kowalczyk
    • 2
  • Monica Musso
    • 3
    • 4
  1. 1.Departamento de Ingeniería Matemática and CMMUniversidad de ChileSantiagoChile
  2. 2.Department of Mathematical SciencesKent State UniversityKentUSA
  3. 3.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly
  4. 4.Departamento de MatemáticaPontificia Universidad Catolica de ChileMaculChile

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