My research paper talks about the downsides of love, and argues how it is a source of suffering. I am using British literature, specifically Shakespeare's King Lear, to support my argument. How does love cause conflict in Lear's life?

The entire conflict in King Lear is set off by Lear giving his daughters a test of love, and assuming that their eloquence in describing their love for him matches the affection that they actually feel for him.
Lear is aging, and he attempts to avoid conflict after his death by publicly declaring how he will divide up his kingdom. He has three daughters, and he tells them:

Tell me, mydaughters—Since now we will divest us both of rule,Interest of territory, cares of state—Which of you shall we say doth love us most,That we our largest bounty may extendWhere nature doth with merit challenge (1.1.52-57).


Essentially, Lear has said that he will give the biggest portion of his kingdom to the daughter who loves him most, and is therefore most deserving of this gift. Lear asks his daughters to describe their love for him, and he will decide who gets the biggest piece of the kingdom after hearing what they have to say.

His first daughter, Goneril, gives a hyperbolic description of her love, saying that she loves him


more than word can wield thematter,Dearer than eyesight, space, and liberty,Beyond what can be valued, rich or rare,No less than life, with grace, health, beauty, honor;As much as child e’er loved, or father found;A love that makes breath poor, and speech unable.Beyond all manner of so much I love you (1.1.60-67).


His second daughter, Regan, is equally ridiculous, saying,



I professMyself an enemy to all other joysWhich the most precious square of sensepossesses,And find I am alone felicitateIn your dear Highness’ love (1.1.79-84).


Lear's third daughter, Cordelia, however, only says

I love your Majesty / According to my bond, no more nor less (1.1.101-102).



Cordelia says that she loves Lear as much as she owes him, because he is her father. Lear is outraged and thinks that Cordelia is cold and unfeeling, but only Cordelia has spoken the truth about her feelings.

Lear divides up his kingdom based on the flattery of his first two daughters, but their affection is not true. Neither one of them truly loves him but only cares about herself. Goneril and Regan spend the rest of the play belittling Lear and destroying his kingdom in a series of escalating conflicts. Cordelia was the only one who actually had true affection for her father, and she loved him as a daughter should. However, Lear confuses flattery and words with real love, and by the time he recognizes this it is too late. Cordelia is killed in the string of events that follow from the division of the kingdom, and Lear dies of grief.
https://www.folgerdigitaltexts.org/html/Lr.html