I think because of the type of character he is. I think it would destroy him if Elena chose Stefan over him.

It will be like his obsession over Katherine all over again, but this time he will be way in too deep. Damon won't quit fighting for Elena till she herself tells him to stop or till she stops breathing. He will possibly do ANYTHING to make sure he will be the only one she loves, not including his brother, this time.

Completely agree! ^^

I don't get the question. Give her up as in let her go? If yes then...
I think he's the complete opposite type of guy actually. It takes a lot for him to really care about someone or love him/her but once he does it's so hard to make him stop loving them. Like with Katherine. He loved her for a whole eternity and it's only now that he seems to be letting her go after he found out she's a bitch. lol.
I think it'll be the same with Elena. But If he does think she'll be happier without him, he will let her go.

Because he loves her...He loved Katherine too and he couldn't let her go for 145 years...And I think he loves Elena more...so I don't think so.

He loves her and he is the type of guy who doesn't give up to someone he loves. It's also got to do with his attitude.. He never stops fighting for what he wants and he will keep doing that till she is his! <33 I love that about him. :)

because Damon NEVER gives up easily. Especially when it comes to the woman he loves! I'm sure he will do everything to win Elena's love.

Because deep down Damon has feelings (REAL & TRUE) 4 Elena & i fink he will slowly start to reliase that as the show goes on (he will ave more feelings 4 Elena than Katherine even though he loved Katherine at the begining & that she was the 1 who made him a vampire). Near the end of season 1 it proved that katherine didn't love Damon as much as he 4ght she did by not returning back 2 him. Poor Damon he 4ght she was still in the tomb but she wasn't :(. He knows now that Katherine didn't really love him but if he finks he has a chance to win over Elena's heart i'm pretty sure he will do anything 4 her. It's a shame that Eleans wiv Stefan thou cuz wheneva there on the scene i cud serviously fall a sleep. I'm sure Damon will go down fighting for Elena xxx

He's just the type to not give up on her. I mean... look at how long he was longing for Katherine ;)

NO! Damon always gets what he wants. But, if it was to protect Elena, Damon would leave because he's GOOD to the bone! <3

He will never give up on her because that is how much he loves her!

^ because he loves her

NOWAY!!!! because when damon falls in love, he falls HARD!!! and as you can see, what he feels for elena, is the kind of love that doesn't come along every day!! he is haed over heels in love with her, and i don't think that's ever going to change!! EVER!!