Differential and Integral Equations

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

Biao Ou

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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


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