  In this paper, we study a general class of Hessian elliptic equations,
including the Monge-Amp\`ere equation, the $k$-Hessian equation and
$p$-Monge-Amp\`ere equations. We propose new additional condition on the
solution and prove Liouville theorem under this assumption. We show that our
general condition covers as special cases numerous sets of assumptions known in
the literature which were tailored for specific equations. Thus we obtain a
significant generalization of multiple isolated Liouville theorems and
conditional interior estimates.
