Recently, great attention has been focused on the study of fractional and non-local operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. Our problem is related to the fractional Yamabe problem. First, we study a non-local problem involving the fractional laplacian, a critical nonlinearity with a non-symmetric weight,and a perturbation in a bounded domain. We show that in the case of a linear perturbation. Next, if the perturbation is nonlinear, we find non-ground-state solutions for the problem.