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

Abstract

In this article, we study the nonperiodic damped vibration problem $$ \ddot{u}(t)+q(t)\dot u(t)-L(t)u(t)+\nabla W(t,u(t))=0, $$ where L(t) is uniformly positive definite for all $t\in \mathbb{R}$, and W(t,x) is either subquadratic or asymptotically quadratic in x as $|x|\to \infty$. Based on the minimax method in critical point theory, we prove the existence and multiplicity of fast homoclinic solutions for the above problem.

Keywords

• Fast homoclinic solutions
• damped vibration problem
• subquadratic
• asymptotically quadratic