Fast homoclinic solutions for damped vibration systems with subquadratic and asymptotically quadratic potentials

Abstract

In this article, we study the nonperiodic damped vibration problem ü(t) + q(t)u̇(t) - L(t)u(t) + ∇W(t, u(t)) = 0, where L(t) is uniformly positive definite for all t ∈ ℝ, and W(t, x) is either subquadratic or asymptotically quadratic in x as |x| → ∞. Based on the minimax method in critical point theory, we prove the existence and multiplicity of fast homoclinic solutions for the above problem.