How bitcoin maintain its network when bitcoin no longer mineable?
Miner's fees.
More or less, whenever we get around to that point, assuming the network had been constantly expanding and growing, we would see Bitcoin at a relatively high value, however the actual economics of it are fuzzy because there would be nearly no minimum value Bitcoin would be sold for. Aside from that this would allow for users to send relatively small fees along with their transactions (maybe smaller than today's if the block sizes are increased and the volume also increases) and yet pay the miners enough for everything to continue operating.