Himizu (movie, japanese)
It deals with severe depression and how one can become crazy because of being lonely and growing up in a not loving family.
Both the main boy and girl deal with it. The depression and loneliness made the boy a loner who's biggest wish is to live a normal life. It made the girl turn into a crazy person who talks to herself and who becomes obsessed with the boy, probably because she felt he was similar.
This movie is really heavy, the actors did in my opinion a really good job.
Han Gong Ju (movie, korean)
Another heavy movie. The girl acts quite rude and also prefers to be alone. The reason why is a mystery until the end. I can't say too much without spoiling. This is also based of a true story and it will make you lose faith in humanity so you might take that into consideration.
Bleak Night (movie, korean)
A story of 3 boys who used to be good friends. The movie starts after the father of one of the boys starts looking for the reason why his son commited suicide. The dad really had not much to do with the whole movie and most of the movie takes place in the past and explains what happened to the former 3 friends. Even though it's about 3 friends you will feel the loneliness when watching the movie.
Yuuki (movie, japanese)
this one is really different from the other 3 as it deals with an illness (not a mental one though). It's a really touching movie and a real tearjerker. Why I recommend this movie is basically because the guy keeps smiling while dying inside to not make others worry and his friends are kinda not there for him because of that.
Dear Friends (movie, japanese)
I think the sypnosis gives enough explanation as to why I think the main girl is really lonely
''Dear Friends follows a high-school student named Rina (played by Keiko Kitagawa) who believes that friends are not necessary and that they can only be used in times of need. Thus, she is unable to maintain a decent relationship with her friends and classmates.''
Rinne no Ame (movie, japanese)
This one deals with a mental illness and also a little bit of loneliness as the brothers are both abandoned by their dad after their mother passed away. So yeah it's basically about 2 brothers and the younger one has a mental illness which makes him have the mind of a young boy and he kinda lives in his own world (im not sure how to explain in english but I guess you understand).
well actually all of these movies are heavy though I would say the first 3 are really heavy
I don't really know a drama. I think Piece dealed a bit with it. You could at least say the male lead wasn't normal and a bit crazy. Also in clone baby all of the side characters dealed with loneliness and, well it's a supernatural drama, so they dealed with that too. Both are really weird and they are japanese even though you said korean series, but I loved them, especially clone baby. Oh and mars deals with traumas, but it's taiwanese