Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent

In the present paper we study the weak lower semicontinuity of a functional containing a critical term and depending on two real parameters. As a consequence of this regularity result we prove the existence of a nontrivial weak solution for two different nonlocal critical equations driven by the fractional Laplace operator. These two existence results were obtained using, respectively, the direct method in the calculus of variations and critical points theory.