I can't change my passport(in my country we keep our name) and I have to go to my country in two weeks.I know I have no problem in the US airport.But what about my country?When I have to come back to the States,in the european airport they will ask me the green card,cause they see I don't have a return-flight(last summer my friend went to Italy and in Milan airport they asked her the green card when she was coming back to the States).So,can I get problems to leave Europe with a different name on my documents?thanks

Buy ticket with the name in your passport. You will need to show green card to immigration when you return to US only.