On the Palais principle for non-smooth functionals

TL;DR

This paper extends Palais' criticality principle to non-smooth, lower semi-continuous functionals invariant under compact Lie group actions, enabling new analysis of PDEs previously intractable with classical methods.

Contribution

It introduces new versions of Palais' principle applicable to non-smooth functionals, broadening the scope of critical point theory for PDEs.

Findings

Extended Palais' principle to non-smooth functionals.

Applied the theory to PDEs with lower semi-continuous functionals.

Enabled analysis of PDEs previously inaccessible with classical methods.

Abstract

If $G$ is a compact Lie group acting linearly on a Banach space $X$ and $f$ is a $G$-invariant function on $X$, we provide new versions of the so-called Palais' criticality principle for $f:X\to\bar\R$, in the framework of non-smooth critical point theory. We apply the results to a class of PDEs associated with functionals which are merely lower semi-continuous and could not be treated by previous versions of the principle.