Monotonicity of positive solutions for nonlocal problems in unbounded domains.

In this paper, we consider the following problem { (− Δ) p s u (x) = f (u) , u (x) > 0 in Ω , u (x) = 0 on R n ∖ Ω , where (− Δ) p s is the fractional p -Laplacian with p ≥ 2 and Ω : = { x = (x ′ , x n) | x n > φ (x ′) } is an unbounded domain. In the case φ ≡ 0 , it reduces to the upper half space. Without assuming any asymptotic behavior of u near infinity, we first develop narrow region principles in unbounded domains, then using the method of moving planes, we establish the monotonicity of the positive solutions. In most previous literature, to apply the method of moving planes on unbounded domains, one usually needed to assume that the solution tends to zero in certain rate near infinity; or make a Kelvin transform, or divide the solution by a function, so that the new solution possesses such an asymptotic decay. Here we introduce a new idea, estimating the singular integral defining (− Δ) p s along a sequence of auxiliary functions at their maximum points. This way, we only require the solutions be bounded. We believe that this new method will become a useful tool in investigating qualitative properties of solutions for equations involving nonlinear nonlocal operators. [ABSTRACT FROM AUTHOR]