Fast homoclinic solutions for damped vibration problems with superquadratic potentials

Abstract

In this paper we investigate the existence and multiplicity of homoclinic solutions for the following damped vibration problem: u + q(t)u-L(t)u + W u (t, u) = 0, (DS) where q : R → R is a continuous function, L ∈ C(R, R n 2) is a symmetric and positive definite matrix for all t ∈ R and W ∈ C 1 (R × R n , R). The novelty of this paper is that, assuming lim |t|→+∞ Q(t) = +∞ (Q(t) = 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.