Non-Nehari Manifold Method for Hamiltonian Elliptic System with Hardy Potential: Existence and Asymptotic Properties of Ground State Solution

This paper is dedicated to studying ground state solution for a class of Hamiltonian elliptic system with gradient term and inverse square potential. The resulting problem engages four major difficulties: one is that the associated functional is strongly indefinite, the second is that, due to the asymptotically periodic assumption, the associated functional loses the Z(N)-translation invariance. The third difficulty we must overcome lies in verifying the link geometry and showing the boundedness of Cerami sequences when the non-linearity is asymptotically quadratic. The last is singular potential mu/vertical bar x vertical bar(2), which does not belong to the Kato's class. These enable us to develop a direct approach and new tricks to overcome the difficulties caused by singularity and the dropping of periodicity of potential. We establish the existence and non-existence results of ground state solutions under some mild conditions, and derive asymptotical behavior of ground state solutions.