摘要：As well known, the existence and nonexistence of solutions for nonlinear
algebraic systems are very important since they can provide the necessary
information on limiting behaviors of many dynamic systems, such as the
discrete reaction-diffusion equations, the coupled map lattices, the
compartmental systems, the strongly damped lattice systems, the complex
dynamical networks, the discrete-time recurrent neural networks, and the
discrete Turing models. In this paper, both the existence of nonzero
solution pairs and the nonexistence of nontrivial or nonzero solutions for
a nonlinear algebraic system will be considered by using the critical point
theory and Lusternik-Schnirelmann category theory. The process of proofs on
the obtained results is simple, the conditions of theorems are also easy to
be verified, however, some of them improve the known ones even if the system
is reduced to the precial cases, in particular, others of them are still
new.
