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We consider the steady-state Boussinesq system in the whole three-dimensionalspace, 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 thepossibly non-existence of $L^p-$solutions for $1\leq p \leq 3$. Then, we usethe more general setting of the $L^{p,\infty}-$spaces to show that weaksolutions and their derivatives are H\"older continuous functions, where themaximum gain of regularity is determined by the initial regularity of theexternal forces and the gravitational acceleration. As a bi-product, we get anew regularity criterion for the steady-state Navier-Stokes equations.Furthermore, in the particular homogeneous case when the external forces areequal to zero; and for a range of values of the parameter $p$, we show thatweak solutions are not only smooth enough, but also they are identical to thetrivial (zero) solution. This result is of independent interest, and it is alsoknown as the Liouville-type problem for the steady-state Boussinesq system.

On the existence, regularity and uniqueness of $L^p$-solutions to the steady-state 3D Boussinesq system in the whole space