# Exercise Jurgen Jost's PDE show that u harmonic and nonnegative is constant

2.5: Let u be harmonic and nonnegative, show that u is constant. (Hint use the previous exercise).

The previous exercise was posted in another question, stated the following.

2.4: Let $$u:B(0,R)\subset \mathbb{R^d}\rightarrow\mathbb{R}$$ be harmonic and nonnegative. Prove the following version of the Harnack inequality: $$\dfrac{R^{d-2}(R-|x|)}{(R+|x|)^{d-1}}u(0)\leq u(x)\leq \dfrac{R^{d-2}(R+|x|)}{(R-|x|)^{d-1}}u(0)$$

I used the poisson's integral for the ball that states the following.

$$u(x)=\dfrac{R^2-|x|}{n\alpha(n)R}\int_{\partial B(0,R)}\dfrac{g(y)}{|x-y|^n}dS(y)$$ and the fact that for $$y\in \partial B(0,R)$$, $$|y|=R$$ and $$|x|-|y|\leq |x-y|\leq |x|+|y|$$ that is $$|x|-R\leq |x-y|\leq |x|+R$$.

Any hints on how to apply 2.4 to 2.5? thanks in advance.

• Jan 15 '19 at 17:17

Just let $$R\to\infty$$. Both sides of the inequality converge to $$u(0)$$.