Fractional Elliptic Problems Research Articles

Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space

We study the nonexistence of solutions for fractional elliptic problems via a monotonicity result which obtained by the method of moving planes with an improved Aleksandrov–Bakelman–Pucci type estimate for the fractional Laplacian in unbounded domain.

Nonexistence results for a class of fractional elliptic boundary value problems

In this paper we study a class of fractional elliptic problems of the form{(−Δ)su=f(x,u)in Ω,u=0in RN∖Ω, where s∈(0,1). We prove nonexistence of positive solutions when Ω is star-shaped and f is supercritical. We also derive a nonexistence result for subcritical f in some unbounded domains. The argument relies on the method of moving spheres applied to a reformulated problem using the Caffarelli–Silvestre extension (Caffarelli and Silvestre (2007) [11]) of a solution of the above problem.