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

Abstract

The main purpose of this paper is to study the following damped vibration problems (1.1) { − 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 | → ∞ where A = [ a i , j ( t ) ] ∈ C ( R , R N 2 ) is an N × N symmetric matrix-valued function, B = [ b i j ] is an antisymmetry N × N constant matrix, F ∈ C 1 ( R × R N , R ) and ∇ F ( t , u ) : = ∇ u F ( 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.