On a class of damped vibration problems with super-quadratic potentials

The main purpose of this paper is to study the following damped vibration problems equation(1){ü(t)+(q(t)IN×N+B)u̇(t)+∇F(t,u(t))=A(t)u(t)−12q(t)Bu(t),a.e.t∈[0,T]u(0)−u(T)=u̇(t)−u̇(T)=0 where T>0T>0, q∈L1(0,T;R)q∈L1(0,T;R) with ∫0Tq(t)dt=0, A(t)=[aij(t)]A(t)=[aij(t)] is a symmetric N×NN×N matrix-valued function defined in [0,T][0,T] with aij∈L∞([0,T])aij∈L∞([0,T]) for all i,j=1,2,…,Ni,j=1,2,…,N, B=[bij]B=[bij] is an antisymmetry N×NN×N constant matrix. By establishing a proper variational set, one existence result and two multiplicity results of non-trivial periodic solutions of (1) are obtained.