Critical points on closed convex sets vs. critical points and applications

The existence of multiple critical points for a locally Lipschitz continuous functional $\Phi$ on a closed convex subset $C$ of a Banach space $X$ is investigated. The problem of finding extra conditions under which critical points for $\Phi$ on $C$ turn out to be critical on $X$ is also addressed. Two applications concerning elliptic variational-hemivariational inequalities are then worked out.