Calculus of Variations and Partial Differential Equations

, Volume 24, Issue 1, pp 47–81

Singular limits in Liouville-type equations

Authors

  • Manuel del Pino
    • Departamento de Ingeniería Matemática and CMMUniversidad de Chile
  • Michal Kowalczyk
    • Department of Mathematical SciencesKent State University
  • Monica Musso
    • Dipartimento di MatematicaPolitecnico di Torino
    • Departamento de MatemáticaPontificia Universidad Catolica de Chile
Article

DOI: 10.1007/s00526-004-0314-5

Cite this article as:
del Pino, M., Kowalczyk, M. & Musso, M. Calc. Var. (2005) 24: 47. doi:10.1007/s00526-004-0314-5

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 \(m \ge 1\)

Copyright information

© Springer-Verlag Berlin/Heidelberg 2005