The effects of concave and convex nonlinearities in some noncooperative elliptic systems

We consider a class of elliptic systems leading to strongly indefinite functionals, with nonlinearities which involve a combination of concave and convex terms. Using variational methods, we prove the existence of infinitely many large and small energy solutions. Our approach relies on new critical point theorems which guarantee the existence of infinitely many critical values of a wide class of strongly indefinite even functionals. Our abstract critical points theorems generalize the fountain theorems of T. Bartsch and M. Willem.