Lie Symmetries and Criticality of Semilinear Differential Systems

TL;DR

This paper explores the concept of criticality in semilinear differential systems through Lie symmetry theory, connecting it with scaling transformations and Noether's identities, and demonstrating compatibility with classical critical exponents.

Contribution

It introduces a Lie symmetry-based definition of criticality for semilinear systems, linking symmetry analysis with critical exponents and identities.

Findings

The new definition aligns with classical critical exponents.

Examples illustrate the compatibility of the symmetry-based approach.

The approach relates criticality to scaling and Noether identities.

Abstract

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.