  In this paper, we study the following fully nonlinear elliptic equations
\begin{equation*} \left\{\begin{array}{rl}
\left(S_{k}(D^{2}u)\right)^{\frac1k}=\lambda f(-u) & in\quad\Omega \\ u=0 &
on\quad \partial\Omega\\ \end{array} \right. \end{equation*} and coupled
systems \begin{equation*} \left\{\begin{array}{rl}
(S_{k}(D^{2}u))^\frac1k=\lambda g(-u,-v) & in\quad\Omega \\
(S_{k}(D^{2}v))^\frac1k=\lambda h(-u,-v) & in\quad\Omega \\ u=v=0 & on\quad
\partial\Omega\\ \end{array} \right. \end{equation*} dominated by $k$-Hessian
operators, where $\Omega$ is a $(k$-$1)$-convex bounded domain in
$\mathbb{R}^{N}$, $\lambda$ is a non-negative parameter,
$f:\left[0,+\infty\right)\rightarrow\left[0,+\infty\right)$ is a continuous
function with zeros only at $0$ and $g,h:\left[0,+\infty\right)\times
\left[0,+\infty\right)\rightarrow \left[0,+\infty\right)$ are continuous
functions with zeros only at $(\cdot,0)$ and $(0,\cdot)$. We determine the
interval of $\lambda$ about the existence, non-existence, uniqueness and
multiplicity of $k$-convex solutions to the above problems according to various
cases of $f,g,h$, which is a complete supplement to the known results in
previous literature. In particular, the above results are also new for
Laplacian and Monge-Amp\`ere operators. We mainly use bifurcation theory,
a-priori estimates, various maximum principles and technical strategies in the
proof.
