  We develop a model of Bose-Einstein condensate dark matter halos with a
solitonic core and an isothermal atmosphere based on a generalized
Gross-Pitaevskii-Poisson equation [P.H. Chavanis, Eur. Phys. J. Plus 132, 248
(2017)]. This equation provides a heuristic coarse-grained parametrization of
the ordinary Gross-Pitaevskii-Poisson equation accounting for violent
relaxation and gravitational cooling. It involves a cubic nonlinearity taking
into account the self-interaction of the bosons, a logarithmic nonlinearity
associated with an effective temperature, and a source of dissipation. It leads
to superfluid dark matter halos with a core-halo structure. The quantum
potential or the self-interaction of the bosons generates a solitonic core that
solves the cusp problem of the cold dark matter model. The logarithmic
nonlinearity generates an isothermal atmosphere accounting for the flat
rotation curves of the galaxies. The dissipation ensures that the system
relaxes towards an equilibrium configuration. In the Thomas-Fermi
approximation, the dark matter halo is equivalent to a barotropic gas with an
equation of state $P=2\pi a_s\hbar^2\rho^2/m^3+\rho k_B T/m$, where $a_s$ is
the scattering length of the bosons and $m$ is their individual mass. We
numerically solve the equation of hydrostatic equilibrium and determine the
corresponding density profiles and rotation curves. Our model has no free
parameter so it is completely predictive. Extension of this model to
noninteracting bosons and fermions will be presented in forthcoming papers.
