Sharp conditions for fully nonlinear equations with $(n-1)$-Hessian matrix: Part II. $k$-Hessian case

We study admissible entire subsolutions and radial boundary blow-up solutions of $S_k^{\frac{1}{k}}[\mathrm{\Delta }uI-D^{2}u]=f(u)+g(u)\mathrm{\mid }Du{\mathrm{\mid }}^q$ for $1\le k<n$, continuing our study of the determinant case $k=n$ in Part I. Under structural assumptions on the continuous, non-decreasing coefficients $f$ and $g$, we establish necessary and sufficient conditions for the existence of entire subsolutions when $q<2$. Our results cover the non-Lipschitz case $0<q<1$ with a possibly vanishing zero-order term, the gradient-singular case $q<0$, and sign-changing gradient coefficients. We also characterize finite-radius blow-up of radial solutions and derive two-sided estimates for the blow-up radius, as well as pointwise upper bounds for their profiles in terms of inverse Keller–Osserman functions.