On Palais’ principle for non-smooth functionals

If GG is a compact Lie group acting linearly on a Banach space XX and ff is a GG-invariant function on XX, we provide some new versions of the so-called Palais’ principle of symmetric criticality for f:X→R̄, in the framework of non-smooth critical point theory. We apply the results to a class of quasi-linear PDEs associated with invariant functionals which are merely lower semi-continuous and thus could not be treated by previous non-smooth versions of the principle in the literature.