Infinitely many solutions for diffusion equations without symmetry

We consider the following diffusion system: {∂tu−△xu+b(t,x)∇xu+V(x)u=Hv(t,x,u,v),−∂tv−△xv+b(t,x)∇xv+V(x)v=Hu(t,x,u,v)∀(t,x)∈R×RN, which is an unbounded Hamiltonian system in L2(R×RN,R2m)L2(R×RN,R2m), z:=(u,v):R×RN→Rm×Rmz:=(u,v):R×RN→Rm×Rm, b∈C(R×RN,RN)b∈C(R×RN,RN), V∈C(RN,R)V∈C(RN,R) and H∈C1(R×RN×R2m,R)H∈C1(R×RN×R2m,R). Suppose that H,bH,b and VV depend periodically on tt and xx, and that H(t,x,z)H(t,x,z) is superquadratic in zz as |z|→∞|z|→∞. Without a symmetry assumption on HH, we establish the existence of infinitely many geometrically distinct solutions via a variational approach.