  In this paper we fully characterize the sequentially weakly lower
semicontinuity of the parameter-depending energy functional associated with the
critical Kirchhoff problem. We also establish sufficient criteria with respect
to the parameters for the convexity and validity of the Palais-Smale condition
of the same energy functional. We then apply these regularity properties in the
study of some elliptic problems involving the critical Kirchhoff term.
