[Math] Maximum principle for harmonic functions in unbounded domains

Question

We demonstrated the weak maximum principle for harmonic functions
in bounded domains, proving it first considering the case u
subharmonic, then approximating in this way:
choose $v(x)=x_1^2-M$ so that $\Delta v>0$ and $v<0$.
Consider $u_{\epsilon}(x)=u(x)-\epsilon v(x)$ which is
subharmonic so the principle holds, and taking the limit for
$\epsilon$ to 0 we obtain also the case u harmonic. This proof
depends essentially in the fact that the domain is bounded so we
can choose a negative subarmonic $v$. Can we prove it also if
the domain is not bounded? Of course if it is included in a strip
we can, but otherwise? So the question moves essentially to the existence
of a negative subharmonic function, which i couldn't find..

Answer 1

The key term is the Phragmén-Lindelöf principle. The Wikipedia article talks only about the holomorphic functions. For subharmonic functions, see these notes (which seem to have origin in Potential theory in the complex plane by Ransford).