Conventional mortgage on a undervalued home

[Luke G.]

I have a lead on a house through my mailing campaign that my wife and I are considering purchasing for ourselves.  What I am curious about is the financing.

Say the agreed purchase price is $400k and house appraises for $500k.  Will the lender consider loaning on $500k and we keep the difference or is that unlikely?

If above option does not work, would it be beneficial to purchase the house with our cash and rest with hard/private money and then shortly after the cash purchase get a conventional mortgage on it?

Thank you!