We consider the semi-linear elliptic PDE driven by the fractional


(-\Delta)^s u=f(x,u...
...R}^n\backslash\Omega$.} \\

An $L^{\infty}$ regularity result is given, using De Giorgi-Stampacchia iteration method.

By the Mountain Pass Theorem and some other nonlinear analysis

methods, the existence and multiplicity of non-trivial solutions for

the above equation are established. The validity of the Palais-Smale

condition without Ambrosetti-Rabinowitz condition for non-local

elliptic equations is proved. Two non-trivial solutions are given

under some weak hypotheses. Non-local elliptic equations with

concave-convex nonlinearities are also studied, and existence of at

least six solutions are obtained.

Moreover, a global result of Ambrosetti-Brezis-Cerami type is given,

which shows that the effect of the parameter $\lambda$ in the

nonlinear term changes considerably the nonexistence, existence and

multiplicity of solutions.