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

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

Jesus Ildefonso Diaz, Jesus Hernandez

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.

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Jesús Ildefonso Díaz
Instituto de Matemática Interdisciplinar
Universidad Compluense de Madrid
28040 Madrid, Spain
Jesús Hernández
Departamento de Matemáticas
Universidad Autónoma de Madrid
28049 Cantoblanco, Madrid, Spain

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