  In this paper, we propose a gradient type term for the $k$-Hessian equation
that extends for $k>1$ the classical quadratic gradient term associated with
the Laplace equation. We prove that such as gradient term is invariant by the
Kazdan-Kramer change of variables. As applications, we ensure the existence of
solutions for a new class of $k$-Hessian equation in the sublinear and
superlinear cases for Sobolev type growth. The threshold for existence is
obtained in some particular cases. In addition, for the Trudinger-Moser type
growth regime, we also prove the existence of solutions under either
subcritical or critical conditions.
