  We obtain nontrivial solutions of a critical fractional $p$-Laplacian
equation in the whole space and with possibly vanishing potentials. In addition
to the usual difficulty of the lack of compactness associated with problems
involving critical Sobolev exponents, the problem is further complicated by the
absence of a direct sum decomposition suitable for applying classical linking
arguments. We overcome this difficulty using a generalized linking construction
based on the ${\mathbb Z}_2$-cohomological index.

Analysis of PDEs