Is it possible to construct a $\bf{harmonic}$ function $u:\mathbb{R}^N\to \mathbb{R}$ satisfying:
I - There exist a sequence $x_n\in \mathbb{R}^N$ such that $|x_n|\to\infty$ and $u(x_n)\to\infty$,
II - There exist a sequence $y_n\in\mathbb{R}^N$ such that $|x_n-y_n|\leq\delta_n$, where $\delta_n\to 0$ and $u(y_n)\to 0$?
I think that such function does not exist, but I was unable to prove it.
Thank you.
The function $z\mapsto \sin(z^2)$ is an entire function on $\mathbb C$, so its real part is a harmonic function $\mathbb C\to\mathbb R$. Along the real axis it oscillates between $-1$ and $1$, more and more rapidly as you move away from $0$. So taking $x_n=\sqrt{(2n+\frac12)\pi}$ and $y_n=\sqrt{(2n-\frac12)\pi}$ seems to give what you were looking for.