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

{−¨u(t) − b˙u(t) + a(t)u(t) = ∇f (t, u(t)) a.e. t ∈ r

u(t) → 0, ˙u(t) → 0 as |t| → ∞ (1.1)

where a = [ai,j(t)] ∈ c(r, rⁿ²) is an n ×n symmetric matrix-valued function, b = [bij] is an antisymmetry n×n constant matrix, f ∈ c¹(r×rⁿ , r) and∇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.