## Differential and Integral Equations

- Differential Integral Equations
- Volume 9, Number 5 (1996), 1157-1164.

### 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 $p<n/(n-2).$

#### Article information

**Source**

Differential Integral Equations, Volume 9, Number 5 (1996), 1157-1164.

**Dates**

First available in Project Euclid: 6 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367871536

**Mathematical Reviews number (MathSciNet)**

MR1392100

**Zentralblatt MATH identifier**

0853.35045

**Subjects**

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations

Secondary: 31B05: Harmonic, subharmonic, superharmonic functions 35J25: Boundary value problems for second-order elliptic equations

#### Citation

Ou, Biao. Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition. Differential Integral Equations 9 (1996), no. 5, 1157--1164. https://projecteuclid.org/euclid.die/1367871536