On the Palais principle for non-smooth functionals

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.