Existence of solutions for a nonlinear algebraic system with a parameter

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 LusternikSchnirelmann 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.