Fast homoclinic solutions for damped vibration problems under local conditions

In this paper, we study the fast homoclinic solutions for the following damped vibration problems $\ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0$ , $\forall t \in \mathbb{R}$ , where W only satisfies some local conditions with respect to both t and x. We get infinitely many fast homoclinic solutions for the above system when L is unnecessarily coercive at infinity and W is superquadratic near the origin. Subsequently, assuming that L satisfies $$ \liminf _{|t|\to +\infty}[|t|^{\xi}\inf _{|x|=1}(L(t)x,x)]>0 $$ for some constant $\xi <0$ and W admits a variety of growth conditions near the origin with respect to x, we establish a new compact embedding theorem and obtain infinitely many fast homoclinic solutions via the variant symmetric mountain pass lemma. Our results significantly generalize and improve the previous related ones.