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

Let R + n be the n-dimensional upper half Euclidean space, and let α be any even number satisfying 0 < α < n . In this paper, we study the integral equation on the half-space R + n (0.1) u ( x ) = ∫ R + n ( 1 | x − y | n − α − 1 | x ⁎ − y | n − α ) u p ( y ) d y , u ( x ) > 0 , x ∈ R + n , where x ⁎ = ( x 1 , … , x n − 1 , − x n ) 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 ) α 2 u = u p , in R + n ; u = ( − Δ ) u = ⋯ = ( − Δ ) α 2 − 1 u = 0 , on ∂ R + n , where α is an even number.