High energy positive solutions for a coupled Kirchhoff system

In this paper, we consider the coupled Kirchhoff system in subcritical case and critical case. For subcritical case, we first study the least energy of coupled Kirchhoff system and the limit system by using the Nehari manifold method and exclude the existence of semi-trivial solutions. Then we improve the global compactness lemma of coupled Kirchhoff system to overcome the loss of compactness. Combining with the linking theorem and topological degree, we construct a Palais-Smale sequence in the high energy level and prove the existence of high energy positive solutions. Moreover, the results of the subcritical case are extended to the critical case.