Can you as RAW have a potion with concentration duration and let the potion maintain concentration of the spell? So that it does not end at sunset or when I go to sleep?
No, you can't, but probably not for the reason you're thinking. As per page 99 of the core rules, "effects left to their own concentration start to wear off at sunset and sunrise."
As for "maintains Concentration" potions and digestion in general... I'm pretty sure your stomach would do too much damage to the "item," but I don't know the medieval paradigm logic on digesting food and drink.
You can create a "potion" (a one-use item) with D: Conc. and the added effect "maintains Concentration". However, you have to briefly concentrate every sunrise and sunset on the effect to "maintain" it (just like with a normal item). The first dusk or dawn you fail to perform this brief routine, the effect ends. It could be argued that this does not work if the form of the single-use item is perishable (e.g. a potion that you drink and is then "digested". Just give it another form (a small charm or something) or handwave the "potion" form as a cosmetic thing that should have no negative impact.
Now, there's a far tricker thing you may, or may not do, depending on how you read the rules for "constant" effects. By the rules, you can create a constant effect by instilling a D:Sun, 2uses/day, Environmental trigger effect. Almost everyone I know reads this as saying that the effect is renewed every day, twice a day, but in a seamless fashion that makes it look as if it's constantly maintained. This is also explicit in the rules for "constant effects" as Faerie powers in RoP:F. If you also read it this way, stop reading, nothing else to see here 
However, David Chart has allegedly stated in a message to another member of the forum that in his view these "constant" effects are really only "cast once", and the D:Sun, 2 uses/day and Environmental Trigger is just the "price" of the constant effect -- not a description of how it really works, but just how much you should "pay" for it in terms of effect levels. Indeed, in one of the books (can't remember which one!) there's a magic item that modifies such a constant effect with a Muto Vim power, and the Muto Vim power only has 1use/day rather than 2, on the ground that it only needs to be cast once in the lifetime of the item, when the constant power is first activated. If you subscribe to this view, you can easily make potions that you drink once and are then "permanent", by designing them as Constant effects for the price of D:Sun, 2 uses/day, and Environmental trigger. You can make them as potions, because they effect is only cast once, and they just {\i}cost[i] as if the effect was recast twice a day for D:Sun on an Environmental trigger.
... Except that Charged Items can't be created with uses/day, even on a technicality, so even if you subscribe to the view that the effect is "one and done forever" rather than "seamlessly repeat-cast," (a view which I, incidentally, don't share, though here I am arguing with David Chart, so maybe I should rethink my life decisions) you still can't do it with a charged item. A lesser enchanted perma-potion would work under that interpretation, though, since the potion wouldn't have to survive being ingested for the effect to keep going.
I also disagree with the "one and done forever" view. I believe the majority of the people on the forum do. I also suspect that David Chart came up with it to justify the magic item in the book, and did not think too much on the implications
But if you do subscribe to this view, note that it does work with Charged Items. You are not giving a Charged item uses/day. You are just adding a number of levels to the basic D:Mom effect equal to the cost of D:Sun, 2 uses/day, Environmental Trigger (2 magnitudes + 4 levels). The effect is produced only once, and that's it.
Personally, I don't have a problem with that interpretation, as it's basically the "permanent" duration of pervious editions, but it takes a season to produce....with the understanding that it is affected as a cast spell, rather than as an enchanted item. Thus, PeVi magics can pretty much just destroy them with a single roll.
But for creating (say) doublets of Impenetrable silk to pass out to the grogs, or whatnot? I don't really have a problem with it.
EDIT - and that, as long-term effects, they cause warping as per the Covenent Lab rules. (ie, weird things happen around them every so often, as a consequence of their non-grounded magical nature.)