Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian - Wei, Yuanhong; Su, Xifeng (Calculus of variations and partial differential equations, 2015)
We consider the semi-linear elliptic PDE driven by the fractional Laplacian: ( - Δ ) s u = f ( x , u ) in Ω , u = 0 in R n \ Ω . An L ∞ 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 λ in the nonlinear term changes considerably the nonexistence, existence and multiplicity of solutions.