Duryodhana's and His Son's Specific Enmity Towards Arjuna and His Son

Relationships

1. Balarama was so happy with Duryodhana that he wanted his sister, Subhadra, to marry him. Krishna, however, had other plans. He engineered romance between Subhadra and Arjuna and got her to elope with him. “Look, he is not taking her by force,” said Krishna to pacify his infuriated elder brother, “She is holding the reins of the chariot. She wants to go with him. Who are we to stop her?”
2. Later in the epic, according to folk retellings, Balarama wants his daughter, named variously as Vatsala, or Shashirekha, to marry Duryodhana’s son, Laxman. Krishna stops even this from taking place. He asks Bhima’s son, Ghatotkacha, to help Vatsala elope with Arjuna’s son, Abhimanyu and destroy the wedding plans.

Later when His Son Lakshmana was Killed by Abhimanyu, Duryodhana's instigates Karna to sent a shaft from be hind and brake Abhimanyu's bowstring when he was defending himself against heavy odds. Arjuna's young son was attacked by numerous warriors who surrounded him, when he stood all by himself in the field, deprived of bow and chariot, and in a most cowardly manner, killed him.

Also See
http://saralamahabharat.blogspot.com/2016/04/lakshmana-kumara.html