Fast homoclinic solutions for damped vibration problems with superquadratic potentials

In this paper we investigate the existence and multiplicity of homoclinic solutions for the following damped vibration problem: <($)double over dot> + q(t)(u) over dot - L(t) u + W-u(t, u) = 0, (DS) where q : R -> R is a continuous function, L is an element of C(R,R-n2) is a symmetric and positive definite matrix for all t is an element of R and W is an element of C-1(RxR(n), R). The novelty of this paper is that, assuming lim(vertical bar t vertical bar ->+infinity) Q(t) = +infinity(Q(t) = integral(t)(0) q(s) ds) and L is coercive at infinity, we establish one new compact embedding theorem. Subsequently, supposing that W satisfies the global Ambrosetti-Rabinowitz condition, we obtain some new criterion to guarantee the existence of homoclinic solution of (DS) using the mountain pass theorem. Moreover, if W is even, then (DS) has infinitely many homoclinic solutions. Recent results in the literature are generalized and significantly improved.