Fractional and nonlocal operators of elliptic type arise in a quite natural way in many different contexts. In this paper, we study the existence of solutions for a class of fractional equations, while the nonlinear part of the problem admits some perturbation property. We obtain some new criteria for existence of two and infinitely many solutions, using critical point theory. Some recent results are extended and improved. Several examples are presented to demonstrate the applications of our main results.