Final answer:
The Popes faced a significant problem with the 'lapsi' after persecutions ceased, as many Christians rejected the sacramental acts performed by those who had renounced their faith during times of persecution, leading to the Donatist controversy.
Step-by-step explanation:
The problem that the Popes faced regarding the "lapsi", especially once the persecutions had stopped, was a significant religious and moral dilemma. The lapsi were Christians who had renounced their faith under pressure during the Roman persecutions, particularly during the reigns of Emperors Decius and Diocletian, where the Roman officials saw the refusal to participate in the state cults as treason. Once the persecutions ceased with the Edict of Milan in 313, which granted religious toleration, many Christians could not accept the lapsi, especially as leaders. They believed that any sacramental acts performed by these individuals were invalid, leading to the Donatist controversy.
This issue was more than a theological disagreement; it was a challenge to the spiritual and moral integrity of the Church, questioning the legitimacy of the leadership and sacraments administered by those who had, at one point, renounced their faith. The Church had to address these challenges while it was already struggling with internal power struggles and corruption. This period of the Church's history was marked by strife both within and outside of its ranks, as reflected in the multiple Popes ruling simultaneously and the eventual emergence of reform movements like those led by John Wyclif, Jan Hus, and later Martin Luther.