Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition

Abstract

We prove that all the positive harmonic functions on the upper half space $ \{ x : x= (x_{1}, \cdots, x_{n} ), x_{n} \geq 0 \} $ $ (n \geq 3) $ satisfying the boundary condition $ D_{x_n} (u) = - u^{n/(n-2)} $ are fundamental solutions of the Laplace equation multiplied by proper constants. We also prove that there is no positive harmonic function on the upper half space satisfying the subcritical boundary condition $ D_{x_n} (u) = - u^{p} $ for $p<n/(n-2).$