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

• Published in: Calculus of variations and partial differential equations Vol. 52; no. 1-2; pp. 95 - 124
• Main Authors: Wei, Yuanhong, Su, Xifeng
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2015; Springer Nature B.V
• Subjects: Analysis; Boundary value problems; Calculus of variations; Calculus of Variations and Optimal Control; Optimization; Control; Existence theorems; Mathematical analysis; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Mountains; Nonlinearity; Partial differential equations; Regularity; Systems Theory; Texts; Theorems; Theoretical; 35J60; 47J30
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-013-0706-5

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.