  The present paper is devoted to the study of the well-posedness issue for the
density-dependent Euler equations in the whole space. We establish
local-in-time results for the Cauchy problem pertaining to data in the Besov
spaces embedded in the set of Lipschitz functions, including the borderline
case $B^{\frac Np+1}_{p,1}(\R^N).$ A continuation criterion in the spirit of
the celebrated one by Beale-Kato-Majda for the classical Euler equations, is
also proved. In contrast with the previous work dedicated to this system in the
whole space, our approach is not restricted to the $L^2$ framework or to small
perturbations of a constant density state: we just need the density to be
bounded away from zero. The key to that improvement is a new a priori estimate
in Besov spaces for an elliptic equation with nonconstant coefficients.
