Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems

Abstract

Based on new deformation theorems concerning strongly indefinite functionals, we give some new min–max theorems which are useful in looking for critical points of functionals which are strongly indefinite and satisfy Cerami condition instead of Palais–Smale condition. As one application of abstract results, we study existence of multiple periodic solutions for a class of non-autonomous first order Hamiltonian system ( HS ) z ˙ = J H z ( t , x , z ) , ( t , x ) ∈ R × Ω , where ̇ = d / dt , Ω ⊂ R N N ⩾ 1 is a bounded domain with smooth boundary ∂ Ω , z = ( p , q ) ∈ R M × R M = R 2 M , J = 0 I - I 0 , and H ( t , x , z ) ∈ C 1 ( R × Ω × R 2 M , R ) .