A Vázquez-type strong minimum/maximum principle for partial trace operators

In this paper we study a Strong Minimum Principle of Vázquez type for partial trace operators with gradient terms. More explicitly, given an $ n $-tuple {\bf a} = (a_1, \cdots, a_n) non-negative real numbers a_n>0 $, give sufficient conditions on continuous function H:\mathbb{R}\times\mathbb{R}_0^+\to\mathbb{R} in order viscosity supersolutions of$ \mathcal{P}_{\bf a}(D^2u) H(u, |Du|) $in connected open subsets \mathbb{R}^n that vanish at some point \Omega to identically $. When H depends only the gradient, condition is also necessary. Here belongs class fully nonlinear degenerate elliptic includes min-max operator which defined as sum minimum and maximum eigenvalues Hessian matrix. Under suitable both Maximum Compact Support subsolutions will be investigated. Not does our work cover new wide Hamiltonians not investigated literature, but best knowledge, results are even when reduces standard Laplacian.