Positive solutions of \(\Delta u+u^p = 0\) whose singular set is a manifold with boundary

  • S. Fakhi

DOI: 10.1007/s00526-002-0165-x

Cite this article as:
Fakhi, S. Cal Var (2003) 17: 179. doi:10.1007/s00526-002-0165-x

Abstract.

The aim of this paper is to prove the existence of weak solutions to the equation \(\Delta u+u^p = 0\), with \(n \geq 4\), which are positive in a domain \(\Omega \subset \mathbb{R}^n\) and which are singular along a k-dimensional submanifold with smooth boundary. Here, the exponent p is required to lie in the interval \([\frac{n-k}{n-2-k},\frac{n-k+2}{n-2-k})\), where \(1 \leq k < n-2\) is the dimension of the singular set. In the particular case where \(p = \frac{n+2}{n-2}\) and \(\Omega = \mathbb{R}^n\), solutions correspond to solutions of the singular Yamabe problem.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • S. Fakhi
    • 1
  1. 1.C.M.P.XII Départment de Mathématique Université Paris 12, 61, avenue de Gal de Gaulle, 94010 Créteil Cedex, France (e-mail: fakhi@univ-paris12.fr) FR