Abstract

We are concerned in this survey with singular solutions to semi-linear elliptic problems. An example of the type of equations we are interested in is the Gelfand-Liouville problem -Δu = λeu on a smooth bounded domain Ω of ℝN with zero Dirichlet boundary condition. We explore up to what degree known results for this problem are valid in other situations with a similar structure, with emphasis on the extremal solution and its properties. Of interest is the question of identifying conditions such that the extremal solution is singular. We find that, in the problems studied, there is a strong link between these conditions and Hardy-type inequalities. © 2008 Elsevier B.V. All rights reserved.