A Liouville-type theorem for an integral equation on a half-space R+n

Let R+n be the n-dimensional upper half Euclidean space, and let α be any even number satisfying 0<α0,x∈R+n, where x⁎=(x1,…,xn−1,−xn) is the reflection of the point x about the ∂R+n. We use the moving planes method in integral forms introduced by Chen-Li-Ou to establish a Liouville-type theorem for the integral equation (0.1), which is closely related to the higher-order differential equation with Navier boundary conditions(0.2){(−Δ)α2u=up,in R+n;u=(−Δ)u=⋯=(−Δ)α2−1u=0,on ∂R+n, where α is an even number.