A Freudian analysis of the relationships among Eva, Franklin, and their son Kevin would focus on the parents’ dynamic and the son’s psychosexual development.
Kevin does not seem to develop along the same track as a normal child. A major plot point that serves as evidence of this is his resistance to potty training. Kevin almost seems to enjoy forcing his mother to change diapers well beyond the appropriate age. A Freudian would suggest that this behavior is the result of a prolonged fixation on the anal stage of psychosexual development. This is one indication that something is amiss in Kevin’s psyche.
Another underlying issue is that Eva was a sexually liberated, free spirit before she met Franklin. A Freudian might say she feels repressed or trapped within the confines of marriage. This causes her to resent her first child as an emblem of this entrapment.
In turn, Kevin intuits his mother’s resentment from the beginning, which hinders the normal maternal bonding process. As a result, both Eva and Kevin fail to connect emotionally. This causes Kevin to seek Eva’s attention in other ways, namely, by intentional misbehavior, as he matures. Eva then interprets this behavior as evidence of her son’s sociopathic nature rather than as a desperate plea for attention. This only deepens the divide between mother and son. By adolescence, Kevin’s behavioral patterns are fixed.
Franklin is a sort of patsy when it comes to Kevin. Kevin performs normalcy in front of his father, often being overtly affectionate and mild mannered as a display to anger his mother. A Freudian would suggest that Kevin does this because of an Oedipal desire for Eva. Since Eva is emotionally distant, Kevin vents his latent sexual frustrations in his treatment of Franklin. Freudians would say that this is Kevin’s attempt to make his mother jealous, thereby gaining his mother’s favor. Furthermore, Franklin is a sort of weakened male figure in the sense that he displays few traditionally masculine traits. Kevin interprets Franklin as inferior, a fact that further frustrates him when it comes to his mother.
The failure to resolve Kevin’s Oepidal complex stems from Eva’s resentment of her son, her preference for a perceived “weaker” male, and his stunted psychosexual development.
This also explains why he murders his father but not his mother; it is more than just coincidence. He wants to isolate her so that she is then forced to give him the maternal attention he craves. He ultimately succeeds because he is now imprisoned, and Eva’s guilt over what happened compel her to visit him regularly.