Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian

Abstract

We consider the semi-linear elliptic PDE driven by the fractional Laplacian: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} (-\Delta )^s u=f(x,u) &{} \hbox {in }\Omega , \\ u=0 &{} \hbox {in }\mathbb {R}^n\backslash \Omega .\\ \end{array} \right. \end{aligned}$$ 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.