How does Juliet deal with challenges in the play? Please provide quotations for support.

Juliet tends to deal with challenges by following her heart.  She knows that her family would not approve of her feelings for Romeo, but she pursues those feelings nonetheless.  She says, "O Romeo, Romeo, wherefore art thou Romeo? / Deny thy father and refuse thy name, / Or if thou wilt not, be but sworn my love, / And I'll no longer be a Capulet" (2.2.36-39).  She hopes that Romeo will be willing to stand...

Juliet tends to deal with challenges by following her heart.  She knows that her family would not approve of her feelings for Romeo, but she pursues those feelings nonetheless.  She says, "O Romeo, Romeo, wherefore art thou Romeo? / Deny thy father and refuse thy name, / Or if thou wilt not, be but sworn my love, / And I'll no longer be a Capulet" (2.2.36-39).  She hopes that Romeo will be willing to stand up to his father and even relinquish his family name and honor; if he will not, she is willing to give up her own.  Thus, she ranks the demands of her heart far above anything like family pride; any challenge her family would present to their marriage is nullified by her lack of concern for it.

After Romeo kills Tybalt, Juliet is unsure how to respond.  She knows that she should love her husband, but she also knows that she should hate the man who killed her cousin; the fact that both are the same man leaves her confused.  She wonders, "wherefore, villain, didst thou kill my cousin? / That villain cousin would have killed my husband. / Back, foolish tears, back to your native spring; / [....] / My husband lives, that Tybalt would have slain, / And Tybalt's dead, that would have slain my / husband" (3.2.112-117).  She first calls Romeo a villain for killing Tybalt, then she calls Tybalt a villain for wanting to kill Romeo.  Finally, she vows to stop her crying because her husband yet lives, and the man who would have him dead is dead himself.  Again, she chooses her heart over family obligation, nullifying the challenge presented by Romeo's murder of her kinsman.

When presented with her father's decision to marry Juliet to the County Paris, Juliet once again follows her heart to deal with the challenge.  Risking all her parents' displeasure, she tells her mother, "Now, by Saint Peter's Church, and Peter too, / [Paris] shall not make me there a joyful bride!" (3.5.121-122).  She flat out refuses to follow their orders, and when her father makes her desperate, she devises a reason to go to Friar Lawrence.  She will kill herself if he has no way to help her, and so she chooses to fake her own death -- a frightening prospect in itself -- and follows her heart to be with Romeo rather than marry a man she does not love.