Profinite completions I call a profinite group $G$ Noetherian, if evrey ascending chain of closed subgroups is eventually stable. A standart argument shows that every closed subgroup of a Noetherian profinite group is finitely generated.
A profinite group $G$ is called just-infinite if every nontrivial $M \lhd_c G$ is open.
Let $K$ be a profinite Noetherian just-infinite group. Must $K$ be the profinite completion of some residually finite group $R$?
 A: The answer is yes in general.
Since $K$ is finitely generated, by the Nikolov-Segal theorem it coincides with its own profinite completion. So you simply may take $R=K$.
Perhaps, you also want $R$ to be finitely generated. In this case the answer is no. 
If $K$ is a non-soluble p-adic analytic pro-$p$  group, then it has polynomial subgroup growth. However, finitely generated groups of polynomial subgroup growth are virtually soluble (see the book "Subgroup Growth" of Lubotzky and Segal).
A: It is a completely open problem whether Noetherian pro-$p$ group has finite rank, i.e., there is a bound on the number of generators of closed subgroups, which is equivalent to being $p$-adic analytic. As one can classify all just-infinite $p$-adic analytic pro-$p$ group if indeed any Noetherian pro-$p$ group is $p$-adic analytic, then all one needs to do for pro-$p$ groups is to go over the list. My gut feeling would be that the answer for $p$-adic analytic pro-$p$ groups is yes, but I am not 100% sure.     
