An 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 in the
nonlinear term changes considerably the nonexistence, existence and
multiplicity of solutions.