As mentioned in the comments, here is the text found in the footnotes in the following link https://ohr.edu/5195#_edn16;
There is a machlokes in the Poskim about this.
There are those who permit it as they maintain that it is an Issur Drabbanan and is permitted for beauty purposes. These poskim include Rav Ovadiah Yosef (Taharas HaBayis vol. 3, Dinei Chatzitza 8, pg. 29 - 34), Rav Avigdor Nebenzahl (cited in Taharas HaBayis ibid.), Rav Ezra Batzri (Techumin vol. 10, pg. 282; author of Shu”t Shaarei Ezra) and the Palgei Mayim (of Antwerp; Shu”t vol. 2, 52). Rav Matis Deutsch (Shu”t Nesivos Adam vol. 1: 43) is inclined to permit it for beauty purposes as well, but concludes that most authorities do not accept this reasoning.
Most Poskim disagree and do not permit, except perhaps for a medical case of Kavod Habrios - Including Rav Yosef Shalom Elyashiv (cited in Techumin vol. 18, pg. 114), Rav Yisrael Yaakov Fischer (ibid.), Rav Shmuel Halevi Wosner (Shu”t Shevet Halevi vol. 10, 137; who adds that it should be forbidden due to the ‘srach issur’ of tattooing, as well as it being a ‘maaseh shachatz’), Rav Shlomo Zalman Auerbach (cited in Nishmas Avraham vol. 2 -Y”D 180, pg. 132 s.v u’lechorah; who maintains that in a similar case, when the prohibition was Derabbanan, Rav Shlomo Zalman only permitted it to correct an actual blemish, and not for beauty purposes), Rav Chaim Kanievsky (cited in Shu”t Nesivos Adam ibid., 24), the Mishpetei Uziel (Shu”t, new edition vol. 2, Y”D 22: 3, pg. 89; who, in a similar case, only permitted for medical reasons), the B’tzeil HaChochma (ibid., who, in a similar case only permitted for medical need, extenuating circumstances, or b’makom Mitzvah), the Lehoros Nosson (Shu”t ibid.; who maintains that we should pasken each of these machlokesim lechumra, as if they were all Deoraysa), the Shraga HaMeir (Shu”t vol. 8: 44 and 45; who only permits for medical need), the Rivevos Efraim (responsum published in Shu”t Shav V’Rafa vol. 1, pg. 156 – 157; who only permits for medical need), the Megillas Sefer (on Orach Chaim and Yoreh Deah, 16), the Shav V’Rafa (of Holland; Shu”t vol. 1: 45, who only permits for medical need), and Rav Baruch Shraga (Techumin vol. 18, pg. 110 – 114; who only permits for medical need). Rav Asher Weiss (the renowned Minchas Asher) recently told this author (and more extensively in the recent Shu”t Minchas Asher vol. 2: 56) that it is very problematic to rule leniently and allow permanent makeup unless there is medical need or extenuating circumstances such as ‘Kavod HaBrios’.