Nonlocal critical elliptic equations in homogeneous fractional Sobolev spaces

We prove new multiplicity results for some nonlocal critical growth elliptic equations in homogeneous fractional Sobolev spaces. The proofs are based on an abstract critical point theorem based on the ${\mathbb{Z}}_2$-cohomological index and on a novel regularity result for fractional $p$-Laplacian equations as well as on some compact embeddings.