Abstract: We investigated the existence of solutions for tensor variational inequality problems by using convex analysis. Firstly, we gave a necessary condition for the solution set of tensor variational inequality problems to be empty. Secondly, when the tensor was  positive definite on the asymptotic cone of the set, we proved that the solution set of the tensor variational inequality problems was nonempty and compact, and gave several coercivity conditions  for the solution set of tensor variational inequality problems to be nonempty and compact, and gave a necessary condition for the solution set of tensor variational inequality problems to be nonempty and compact.

Key words: tensor variational inequality, nonemptiness and compactness, asymptotic cone