Application of local linking to asymptotically linear wave equations with resonance II
Mieko Tanaka · SUT Journal of Mathematics · 2004
Existence of a time-periodic solution to a non-linear wave equation with resonance is established by a variational method. We consider the 2π-periodic weak solution to a wave equation □u(x,t)=h(x,t,u(x,t)) of space dimension 1, where h(x,t,ξ) is asymptotically linear in ξ both as ξ→0 or ξ→∞, with the co-efficient as ξ→∞ belonging to σ(□). It was proved that there are some cases, where the difference of h(t,x,ξ) from its linear approximation is not bounded, that guarantee the existence of a non-trivial weak solution. In this paper, we show that the restriction in our previous result imposed on these co-efficients can be further relaxed.