Necessary and sufficient conditions of entire sub-solutions for a (k1, k2)-type Hessian systems with gradient terms

Abstract

In this paper, we aim to discuss a class of (k1, k2)-type Hessian system with gradient terms. In the case of k1 = k2 = 1 and 2 ≤ k1, k2 ≤ n, we obtain a sufficient and necessary condition for the existence of the entire admissible sub-solution of the system according to the value range of different parameters, which is also called the generalized Keller–Osserman condition. Based on this, we also discuss the conditions of existence and non-existence of the entire sub-solution, respectively. Finally, we extend the nonlinear terms to the degenerate case and consider the condition of the existence of the positive sub-solution for the above system.