Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented $k$-Hessian Type Equations in Bounded Domains

We prove the existence of admissible subsolutions to the Dirichlet problem for symmetric augmented $k$-Hessian type equations. An important sufficient condition is the uniform $(k-1)$-$A$-convexity of the domain $\mathrm{\Omega },$ where $A(x,z,p)$ is the augmented symmetric matrix appearing in the equation. This condition was originally introduced by F. Jiang, N. S. Trudinger, and X.-P. Yang and we have chosen a special their case. The structural conditions on the matrix $A(x,z,p)$ include its growth with respect to the variables $z$ and $p,$ particularly requiring that some of its first and second derivatives are sufficiently small in a sufficiently small neighborhood of the boundary. Under certain structural conditions on $A(x,z,p),$ the uniform $(k-1)$-$A$-convexity of $\mathrm{\Omega }$ is also a necessary condition for the existence of admissible subsolutions of the equation in a neighborhood of the boundary. Our results extend the classic result by L. Caffarelli, L. Nirenberg, and J. Spruck from the case $A\equiv 0$ to the general case $A\mathrm{\ne }0.$ Our same theorems are valid also for augmented quotient Hessian type equations.