Existence of Solutions for a Nonlinear Algebraic System

As well known, the existence and nonexistence of solutions for nonlinearalgebraic systems are very important since they can provide the necessaryinformation on limiting behaviors of many dynamic systems, such as thediscrete reaction-diffusion equations, the coupled map lattices, thecompartmental systems, the strongly damped lattice systems, the complexdynamical networks, the discrete-time recurrent neural networks, and the discrete Turing models. In this paper, both the existence of nonzerosolution pairs and the nonexistence of nontrivial or nonzero solutions fora nonlinear algebraic system will be considered by using the critical pointtheory and Lusternik-Schnirelmann category theory. The process of proofs onthe obtained results is simple, the conditions of theorems are also easy tobe verified, however, some of them improve the known ones even if the systemis reduced to the precial cases, in particular, others of them are stillnew