Understand that having no EQUITY in your house, was the biggest reason why we had so many foreclosures.
There are programs; fewer than we used to have, and sporatic at best.
Also, understand that as a rule, if you have less than 20% equity, the Lenders require you to have PMI, which insures the Lender in the event you default: This can add a lot to your monthly mortgage.
If you put 10% down, then you will pay PMI until you reach that 20% level, when you can petition the Lender to stop doing that.
Talk to several Lenders; look at their GFE's and compare the numbers.
Good luck and may God bless