In this paper we study some nonlinear elliptic equations in obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is Equation>where , , is a small parameter, , and h is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov-Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.