Alienware adapter problem

Hello! Today I found my laptop has a power problem. When I am using with AC Power Adapter plugged in, (the battery logo at the low righ corner shows "plugged in,not charging") sometimes the screen turns to darker and the logo changes to "using the battery" while the adapter is still in. I tried to adjust the power plug without pull it out but it won't work. Then I found if I pull it out and re-insert it, the power goes back. So I am wondering if it is some connecting problem. If it is, then the problem is from the adapter or the part of the laptop body?

BTW, I bought the laptop five months ago and didn't buy the warranty and only have one year limited warranty. So do I have the right to have Dell repair it for free?

Thank you!