Variational and Topological Methods: Theory, Applications, 
Numerical Simulations, and Open Problems.
Electron. J. Diff. Eqns., Conference 21 (2014), pp. 31-44.

Title: Positive and free boundary solutions to singular nonlinear
       elliptic problems with absorption: an overview and open problems

Authors: Jesus Ildefonso Diaz (Univ. Autonoma de Madrid, Spain)
         Jesus Hernandez      (Univ. Autonoma de Madrid, Spain)

Abstract: 
 We give a survey of recent results and open problems concerning existence
 and multiplicity of positive and/or compact support solutions to some
 semilinear elliptic equations with singular nonlinear terms of absorption
 type. This includes the case of discontinuous (at the origin) nonlinearities,
 which is treated by introducing maximal monotone graphs. Extensions to the
 p-Laplacian are also considered. The one-dimensional case is studied by
 using energy methods.

Published February 10, 2014.
Math Subject Classifications: 35J75, 35J61, 35J92.
Key Words: Singular nonlinearity; positive solution; compact support  solutions;
           free boundary problem; multiplicity; energy methods; absorption; 
           multivalued graph.