Korea was subordinate to China before Japan took it as a colony. The present border was decided between Korea and Chinese Qing Dynasty, and then reconfirmed by Japan and China. The last time that Koreans owned Changbai Mountain was almost at least half a millenium ago. They have nothing to support their claim.

Given the friendship between China and North Korea in 1950s, Mao agreed to give part of the Changbai Mountain to North Korea because Kim Il Cheng claimed that he was born there. China has no land borders with South Korea, so South Korea has no legal right to claim anything regarding Changbai Mountain.