  Prescribing $\sigma_k$ curvature equations are fully nonlinear
generalizations of the prescribing Gaussian or scalar curvature equations.
Given a positive function $K$ to be prescribed on the 4-dimensional round
sphere. We obtain asymptotic profile analysis for potentially blowing up
solutions to the $\sigma_2$ curvature equation with the given $K$; and rule out
the possibility of blowing up solutions when $K$ satisfies a non-degeneracy
condition. We also prove uniform a priori estimates for solutions to a family
of $\sigma_2$ curvature equations deforming $K$ to a positive constant under
the same non-degeneracy condition on $K$, and prove the existence of a solution
using degree argument to this deformation involving fully nonlinear elliptic
operators under an additional, natural degree condition on a finite dimensional
map associated with $K$.
