A semilinear Dirichlet problem involving the fractional Laplacian in R+ n

Abstract

We investigate the Dirichelt problem involving the fractional Laplacian in the upper half-space R + n = x ∈ R n ∣ x 1 > 0 ${\mathbb{R}}_{+}^{n}=\left\{x\in {\mathbb{R}}^{n}\mid {x}_{1}{ >}0\right\}$ : ( − Δ ) s u ( x ) = f ( u ( x ) ) , x ∈ R + n , u ( x ) > 0 , x ∈ R + n , u ( x ) = 0 , x ∉ R + n . \begin{cases}\quad \hfill & {\left(-{\Delta}\right)}^{s}u\left(x\right)=f\left(u\left(x\right)\right),\qquad x\in {\mathbb{R}}_{+}^{n},\hfill \\ \quad \hfill & \qquad u\left(x\right){ >}0,\qquad x\in {\mathbb{R}}_{+}^{n},\hfill \\ \quad \hfill & \qquad u\left(x\right)=0,\qquad x\notin {\mathbb{R}}_{+}^{n}.\hfill \end{cases}. . We prove the positive solutions are monotonic increasing in the x 1-direction assuming u(x) grows no faster than |x| γ with γ ∈ (0, 2s) for |x| large. To start with, we develop a maximum principle on the narrow region. Then we apply a direct method of the moving planes for the fractional Laplacian to derive the monotonicity. As an application of the monotonicity result, we use it to prove nonexistence of bounded positive solutions in R + n ${\mathbb{R}}_{+}^{n}$ for f(u) = u p , p ∈ 1 , n − 1 + 2 s n − 1 − 2 s $p\in \left(1,\frac{n-1+2s}{n-1-2s}\right)$ .