Nonexistence results for a nonvariational system with variable exponents and existence via blow-up

We consider here some systems of nonlinear partial differential equations containing variable exponents in the operator and also in the nonlinear terms. Our aim is to prove some nonexistence and existence results of positive radial solutions.The study of the existence of positive radial solutions for scalar equations containing variable exponents has received some attention in recent years by using variational methods. In this paper, we deal with nonvariational systems, and therefore variational methods do not apply.In this paper, we derive first some nonexistence results, i.e. Liouville type of theorems, for systems of differential equations containing variable exponents, and then we prove the existence of solutions by extending the blow-up method to our system in order to obtain a priori bounds, we finally use topological degree methods to show the existence of positive radial solutions. As far as we know, the systems we study and the results we obtain are new. In this sense, an interesting situation arises as an outcome of the blow up method, and it happens that one obtains a limiting system which does not contain variable exponents, providing in this way a different situation to the known one for the case of constant exponents and $ p $-Laplace operators as in [6].