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 only once within the game: 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 (Brooklet Hill): Level 60 Lanturn, Level 60 Cloyster, Level 60 Araquanid
Captain Kiawe: You can battle Captain Kiawe twice within the game: once during Captain Mina's Trial (Wela Volcano Park), and again in Royal Avenue (after meeting him within his house in Paniola Town) during the Post Game. He is a Fire Type Pokemon User.
Captain Kiawe (Wela Volcano Park): Level 51 Arcanine, Level 51 Talonflame, Level 51 Marowak
Captain Kiawe (Royal Avenue): Level 60 Arcanine, Level 60 Talonflame, Level 60 Marowak
Captain Mallow: You can battle Captain Mallow only once within the game: during Captain Mina's Trial (Lush Jungle). She is a Grass Type Pokemon User.
Captain Mallow (Lush Jungle): Level 51 Trevenant, Level 51 Shiinotic, Level 51 Tsareena
Captain Sophocles: You can battle Captain Sophocles twice within the game: once during Captain Mina's Trial (Hokulani Observatory), and again at the Pokemon League (Championship Challenger) during the Post Game. He is an Electric Type Pokemon User.
Captain Sophocles (Hokulani Observatory): Level 51 Togedemaru, Level 51 Magenezone, Level 51 Golem
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