We consider the nonlinear elliptic PDE driven by the fractional Laplacian with asymptotically linear term. Some results regarding existence and multiplicity of non-trivial solutions are obtained. More precisely, information about multiple non-trivial solutions is given under some hypotheses of asymptotically linear condition; non-local elliptic equations with combined nonlinearities are also studied, and some results of local existence and global existence are obtained. Finally, an $L^{∞}$ regularity result is also given in the appendix, using the De Giorgi-Stampacchia iteration method.