Entire solutions to the harmonic extension of a pseudo-relativistic Hartree equation

This paper is devoted to the study of entire positive solutions of a harmonic equation in \(\mathbb {R}^{n+1}_{+}\) ${\mathbb{R}}_{+}^{n+1}$ with non-local Neumann boundary conditions. We translate the equations into the equivalent integral form. By making use of the moving sphere method, we give a classification of entire positive solutions.