Is it permissible for a young man and a woman to get married if they love each other, despite her family's rejection of him due to his mother's multiple marriages after his father's death, and what should he do?

In Islam, there is no concept known as "love before marriage" due to the corrupting influences it entails. Your desire to marry the girl is legitimate if she possesses good character and religious devotion. Her family should accept you as a husband for their daughter if you observe your religious obligations and abstain from vices, for the Prophet, peace and blessings be upon him, said: "If someone whose religion and character you approve comes to you, then marry him. If you do not do so, there will be tribulation on earth and widespread corruption." A man's mother marrying four men legitimately is not a flaw in her, nor does it relate to her son in terms of his religion and character.