  In this paper we prove the existence of infinitely many nontrivial solutions
for the class of $(p,\, q)$ fractional elliptic equations involving
concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further,
when the nonlinearity is of convex-critical type, we establish the multiplicity
of nonnegative solutions using variational methods. In particular, we show the
existence of at least $cat_{\Omega}(\Omega)$ nonnegative solutions.
