Based on these facts alone, you do not have a lemon law claim as lemon law is warranty law and you received no warranty. Regardless, often times dealers either set a car up to run for a little while, which can be a fraud OR they simply do a poor job of filling out paperwork, which can give you other rights. It's best to find someone local who knows this area of law for a low or no fee consultation.
Find someone good here:
There is no real way to say what you can do without going over all the sales paperwork and listening to the whole story to find how the dealer violated the law and that really means you should talk to a local lemon lawyer near you right away. You can find a list of them at www.USLemonLawyers.com. They don't pay to be listed there and most of them are members of the only national professional group in the country, Naca.net. Thanks for asking and good luck.
This answer is for general purposes only and does not establish an attorney-client relationship. The law in your state may differ and your best answer will always come from a local attorney that you meet with privately. For a Free Online 50 State National List of Consumer Law Lawyers, click on this link (http://tinyurl.com/79ku5jx) and find one near you