Ground State Homoclinic Solutions for Damped Vibration Systems with Periodicity

In this paper, we are concerned with the following periodic damped vibration system \(\begin{aligned} \ddot{u}+q(t){\dot{u}}-L(t)u+\nabla W(t,u)=0. \end{aligned}\) $$\begin{array}{c}\hfill \ddot{u}+q(t)\dot{u}-L(t)u+\mathrm{\nabla }W(t,u)=0.\end{array}$$ Using variational methods and a version of the concentration compactness principle, we study the existence of ground state homoclinic solutions for this system under two different classes of superquadratic conditions weaker than the ones known in the literature. To the best of our knowledge, there has been no work focused in this case.