  We obtain nontrivial solutions to the Brezis-Nirenberg problem for the
fractional $p$-Laplacian operator, extending some results in the literature for
the fractional Laplacian. The quasilinear case presents two serious new
difficulties. First an explicit formula for a minimizer in the fractional
Sobolev inequality is not available when $p \ne 2$. We get around this
difficulty by working with certain asymptotic estimates for minimizers recently
obtained by Brasco, Mosconi and Squassina. The second difficulty is the lack of
a direct sum decomposition suitable for applying the classical linking theorem.
We use an abstract linking theorem based on the cohomological index proved by
Perera and Yang to overcome this difficulty.
