Determine all positive harmonic functions

Asked Apr 29 '21 at 00:47

Active Apr 29 '21 at 05:06

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If u is a positive harmonic function on $\mathbb{R^n}\setminus\{0\}$, show that there exists a,b which is non-negative, such that

$$u(x)=a+b|x|^{2-n}\text{ for all }x\in\mathbb{R^n}\setminus\{0\}$$

Thanks for your concentration. This is a problem from Yau-contest. I try to use Liouville Theorem, however I don't know where to begin.
Any advice will be appreciated!

edited Apr 29 '21 at 05:06

Calvin Khor

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asked Apr 29 '21 at 00:47

Moumou Ye

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$|x|^{2-n}$ is a fundamental solution of Laplace equation in $R^n$ https://math.stackexchange.com/questions/1341370/fundamental-solution-to-the-poisson-equation-by-fourier-transform or https://math.stackexchange.com/questions/4099027/computations-problem-with-reverse-fourier-transform/4099754#4099754 – Svyatoslav Apr 29 '21 at 01:33
the foundamental solution is $\frac{1}{n(2-n)w_{n}|x|^{n-2}}$, however, how to apply? @Svyatoslav – Moumou Ye Apr 29 '21 at 02:26
What's Yau contest? – Calvin Khor Apr 29 '21 at 02:51
One Math Competition for College Students, however this is not important, how to deal with this problem? @Calvin Khor – Moumou Ye Apr 29 '21 at 02:55
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I found the question, 2018 past exam paper, Analysis and Differential Equations Individual, question 6. There is a partial result here – Calvin Khor Apr 29 '21 at 04:24
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I found the full proof. It is quite long, you may check sections 3.9 - 3.14 of this book on Harmonic functions. The result is called Bôcher's Theorem (or rather, it is a corollary of it.) That book cites this paper as the source of the proof. – Calvin Khor Apr 29 '21 at 04:39
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@Calvin Khor, Thank you every much! – Moumou Ye Apr 29 '21 at 05:55

Determine all positive harmonic functions

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