1996 Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition
Biao Ou
Differential Integral Equations 9(5): 1157-1164 (1996). DOI: 10.57262/die/1367871536

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

Citation

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Biao Ou. "Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition." Differential Integral Equations 9 (5) 1157 - 1164, 1996. https://doi.org/10.57262/die/1367871536

Information

Published: 1996
First available in Project Euclid: 6 May 2013

zbMATH: 0853.35045
MathSciNet: MR1392100
Digital Object Identifier: 10.57262/die/1367871536

Subjects:
Primary: 35J65
Secondary: 31B05 , 35J25

Rights: Copyright © 1996 Khayyam Publishing, Inc.

Vol.9 • No. 5 • 1996
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