Is this language regular or not? $L_1=\{a^ku \mid u \in \{a,b\}^* $ and $u$ contains at least $k$ a's, for $k\geq 1\}$.

If it is regular, I haven't found its regular expression or any closure property to prove it.

If not, it seems pumping theorem holds for it...

Also, what about the language where the above "at least" is changed to "at most"?
Take $k=1$. No need to go further, any more $a$ can be absorbed by $u$ without risk.

So you need $u$ to have at least 1 $a$ and only $a$'s or $b$'s. Therefore, $L_1$ is described by $a(a|b)^*a(a|b)^*$ and is regular.

For the side question (at most $k$ $a$'s, let's call that language $L_2$), you can reason in a similar fashion to get $u$ as small as possible. For simplicity let's consider $L_3$ the intersection of $L_2$ with the one denoted by $a^*ba^*$. You have $L_3=\{a^k b a^j, k \geq j\}$. Intersection with its reverse yields $\{a^kba^k\}$ which is not regular, therefore $L_2$ is not regular either since regular languages are closed under intersection and reversal.