Even though the concept of Gym Battles has been replaced in The Alola Region with that of Island Trials, each Trial is still led by a Trial Captain, a trainer that specializes in that given type of Pokemon. And even though you don't battle these Captains during their given Trials - outside of Captain Mina's Trial - you can still battle most of the Trial Captains found within the game at some point or another (primarily during the Post Game).
Trial Captains
Captain Ilima: You can battle Captain Ilima twice within the game: once during Captain Mina's Trial (Hau'oli Cemetery), and again in at the Trainer School (must first meet in his house in Hau'oli City) during the Post Game. He is a Normal Type Pokemon User.
Captain Ilima (Hau'oli Cemetery): Level 51 Gumshoos, Level 51 Smeargle, Level 51 Komala
Captain Ilima (Trainer School): Level 60 Gumshoos, Level 60 Smeargle, Level 60 Komala
Captain Lana: You can battle Captain Lana twice within the game: once during Captain Mina's Trial (Lush Jungle), and again in Brooklet Hill (after defeating her sibilings within their home in Konikoni City) during the Post Game. She is a Water Type Pokemon User.
Captain Lana (Lush Jungle): Level 51 Lanturn, Level 51 Cloyster, Level 51 Araquanid
Captain Lana (Brooklet Hill): Level 60 Lanturn, Level 60 Cloyster, Level 60 Araquanid
Captain Kiawe: You can battle Captain Kiawe only once within the game: during Captain Mina's Trial (Wela Volcano Park). He is a Fire Type Pokemon User.
Captain Kiawe (Wela Volcano Park): Level 51 Arcanine, Level 51 Talonflame, Level 51 Marowak
Captain Mallow: You can battle Captain Mallow only once within the game: in Lush Jungle (after seeing her during the day within the restaurant in Konikoni City) during the Post Game. She is a Grass Type Pokemon User.
Captain Mallow (Lush Jungle): Level 60 Trevenant, Level 60 Shiinotic, Level 60 Tsareena
Captain Sophocles: You can battle Captain Sophocles only once within the game: at the Pokemon League (Championship Challenger) during the Post Game. He is an Electric Type Pokemon User.
Captain Sophocles (Pokemon League): Level 67 Togedemaru, Level 67 Magnezone, Level 67 Electivire, Level 67 Vikavolt, Level 67 Golem
Captain Acerola: You can battle Acerola twice within the game: once during your first attempt at the Pokemon League, and again every time you choose to rematch the Pokemon League. She is a Ghost Type Pokemon User.
Elite Four Acerola (Pokemon League): Level 56 Banette, Level 56 Drifblim, Level 56 Dhelmise, Level 56 Froslass, Level 57 Palossand
Elite Four Acerole (Pokemon League Rematch): Level 66 Banette, Level 66 Drifblim, Level 66 Dhelmise, Level 66 Froslass, Level 66 Palossand
Captain Mina: You will battle Captain Mina only once with the game: during her Island Trial (Seafolk Village). She is a Fairy Type Pokemon User.
Captain Mina (Seafolk Village): Level 51 Mawile, Level 51 Granbull, Level 51 Ribombee
You can actually battle kiawe again post game. If you go to his house, talk to him, then go to the thrifty megamart, he will ask to battle you