Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 053, 17 pages      math-ph/0703071

Lie Symmetries and Criticality of Semilinear Differential Systems

Yuri Bozhkov a and Enzo Mitidieri b
a) Departamento de Matemática Aplicada, Instituto de Matemática, Estatistica e Computação Científica, Universidade Estadual de Campinas - UNICAMP, C.P. 6065, 13083-970 - Campinas - SP, Brasil
b) Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, Via Valerio 12/1, 34127 Trieste, Italia

Received February 01, 2007, in final form March 20, 2007; Published online March 25, 2007

We discuss the notion of criticality of semilinear differential equations and systems, its relations to scaling transformations and the Noether approach to Pokhozhaev's identities. For this purpose we propose a definition for criticality based on the S. Lie symmetry theory. We show that this definition is compatible with the well-known notion of critical exponent by considering various examples. We also review some related recent papers.

Key words: Pokhozhaev identities; Noether identity; critical exponents.

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