The extreme value theorem https://en.wikipedia.org/wiki/Extreme_value_theorem states that a continuous functions $f: \Omega \to \mathbb{R}$ must attain a maximum (and minimum) on $\Omega$ if $\Omega$ is compact. This is not necessarily the case on non-compact (e.g. open) domains.
I wonder wether there is theory on the existence of maxima of continuous functions on non-compact domains $\Omega$. I can only think of somehow proving that the minimum has to be inside a compact subset $\hat{\Omega} \subset \Omega$. But probably there are more sophisticated theorems?
Thanks!