Existence and multiplicity of homoclinic solutions for a class of damped vibration problems

The main purpose of this paper is to study the following damped vibration problems equation(1.1){−ü(t)−Bu̇(t)+A(t)u(t)=∇F(t,u(t))a.e. t∈Ru(t)→0,u̇(t)→0as |t|→∞ where A=[ai,j(t)]∈C(R,RN2)A=[ai,j(t)]∈C(R,RN2) is an N×NN×N symmetric matrix-valued function, B=[bij]B=[bij] is an antisymmetry N×NN×N constant matrix, F∈C1(R×RN,R)F∈C1(R×RN,R) and ∇F(t,u):=∇uF(t,u)∇F(t,u):=∇uF(t,u). By a symmetric mountain pass theorem and a generalized mountain pass theorem, an existence result and a multiplicity result of homoclinic solutions of (1.1) are obtained.