On a Restricted Weak Lower Semicontinuity for Smooth Functional on Sobolev Spaces

We study a restricted weak lower semicontinuity property, which we call the (PS)-weak lower semicontinuity, for a smooth integral functional on the Sobolev space along all weakly convergent Palais-Smale sequences of the functional. By the Ekeland variational principle, the (PS)-weak lower semicontinuity is sufficient for the existence of minimizers under the usual coercivity assumption. In general, this condition is not equivalent to the usual (unrestricted) weak lower semicontinuity condition, but we show that, in certain cases, these two conditions are equivalent and both reduce to the usual convexity or quasiconvexity condition in the calculus of variations.