First price auctions in the asymmetric N bidder case

I consider the first price auction when the bidders' valuations may be differently distributed. I show that every Bayesian equilibrium is an 'essentially' pure equilibrium formed by bid functions whose inverses are solutions of a system of differential equations with boundary conditions. I then prove the existence of an equilibrium. I prove its uniqueness when the valuation distributions have a mass point at the lower extremity of the support. I give sufficient conditions for uniqueness when every valuation distribution is one of two atomless distributions. I establish inequalities between equilibrium strategies when relations of stochastic dominance exist between valuation distributions.