We consider the steady-state Boussinesq system in the whole three-dimensional space, withthe action of external forces and the gravitational acceleration. First, for 3<p <=+infinity we prove the existence of weak L-p-solutions. Moreover, within the framework of a slightlymodified system, we discuss the possibly non-existence of L-p-solutions for 1 <= p <= 3.Then, we use the more general setting of theL(p,infinity)-spaces to show that weak solutions andtheir derivatives are H & ouml;lder continuous functions, where the maximum gain of regularity isdetermined by the initial regularity of the external forces and the gravitational acceleration. Asa 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 parameterp, we show that weak solutions are not only smoothenough, but also they are identical to the trivial (zero) solution. This result is of independentinterest, and it is also known as the Liouville-type problem for the steady-state Boussinesqsystem
