Existence of multiple weak solutions for asymptotically linear wave equations

We consider the problem of multiple existence of 2π2π-periodic weak solutions to wave equations □u(x,t)=h(x,t,u(x,t))+f(x,t)□u(x,t)=h(x,t,u(x,t))+f(x,t) of space dimension 1, where h(x,t,ξ)h(x,t,ξ) is asymptotically linear in ξξ both as ξ→0ξ→0 and as |ξ|→∞|ξ|→∞. It is shown by variational methods that there exist at least three solutions under several conditions on h(x,t,ξ)h(x,t,ξ) if f is sufficiently small. One of the results   reads as follows. Let b≔lim|ξ|→∞∂h/∂ξ(x,t,ξ) and assume that the convergence is uniform with respect to (x,t)(x,t) and that b∉σ(□)b∉σ(□) (non-resonant case). Then the following conditions guarantee the existence of at least three solutions for sufficiently small f  : (a) h(x,t,ξ)-ξ∂h/∂ξ(x,t,0)h(x,t,ξ)-ξ∂h/∂ξ(x,t,0) is non-decreasing (resp. non-increasing) in ξξ, andsup(x,t,ξ)∂h∂ξ(x,t,ξ)max{λ∈σ(□);b>λ}.To obtain these results, we prove that a C1C1-class functional, which is bounded from above and satisfies a condition similar to the linking condition, has at least three critical points under (WPS)*(WPS)* condition.