Could you please explain why "for a bounded domain $\Omega$, $L^p$ space characterize how the function blow up at some point"? Thank you.

Would you mind me asking something not quite relevant to this question? By "a domain $\Omega$", do you mean a connected open set or just an open set? I have seen various usage of the term "domain" in PDE, and it in some cases refers to a connected open set, whereas in other cases refers to an open set, and it is usually the case that the authors are reluctant to explain in advance what they exactly mean by using this term. Similar ambiguity or even (to some extent) abusage seems not rare in PDE. How to you think we should deal with this situation? Thank you.

@mlk Thank you very much! I'll try it.

@mlk Thank you very much! Could you please explain what "the norm induced by $a_{ij}$" means? Forgive me for being stupid.

Also posted in math.stackexchange.com/questions/4133845/…

@GiorgioMetafune Thank you very much! I posted this question because the validity of an outcome in my thesis depends on the monotonicity of $C$, but inspired by your comment, I find it is enough to assure the validity provided $C$ is locally bounded. I will still try to work through the proof and explicitly keep track of the constants involved as it is a great exercise, as is mentioned in the answer math.stackexchange.com/a/4088860/709713.

@leomonsaingeon Sorry there is a typo in the first equation, should have been $\frac{2}{s} u'(s)$. Thank you very much.

@MichaelRenardy Thank you very much!