  We consider the steady-state Boussinesq system in the whole three-dimensional
space, with the action of external forces and the gravitational acceleration.
First, for $3<p\leq +\infty$ we prove the existence of weak $L^p$-solutions.
Moreover, within the framework of a slightly modified system, we discuss the
possibly non-existence of $L^p-$solutions for $1\leq p \leq 3$. Then, we use
the more general setting of the $L^{p,\infty}-$spaces to show that weak
solutions and their derivatives are H\"older continuous functions, where the
maximum gain of regularity is determined by the initial regularity of the
external forces and the gravitational acceleration. As a bi-product, we get a
new regularity criterion for the steady-state Navier-Stokes equations.
Furthermore, in the particular homogeneous case when the external forces are
equal to zero; and for a range of values of the parameter $p$, we show that
weak solutions are not only smooth enough, but also they are identical to the
trivial (zero) solution. This result is of independent interest, and it is also
known as the Liouville-type problem for the steady-state Boussinesq system.
