# Maximum principle for harmonic functions in unbounded domains

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..

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