Existence of multiple weak solutions for asymptotically linear wave equations

✍ Scribed by Mieko Tanaka

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 260 KB
• Volume: 65
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2005.09.022

✦ Synopsis

We consider the problem of multiple existence of 2 -periodic weak solutions to wave equations u(x, t)= h(x, t, u(x, t))+f (x, t) of space dimension 1, where h(x, t, ) is asymptotically linear in both as → 0 and as | | → ∞. It is shown by variational methods that there exist at least three solutions under several conditions on h(x, t, ) if f is sufficiently small. One of the results reads as follows. Let b := lim | |→∞ jh/j (x, t, ) and assume that the convergence is uniform with respect to (x, t) and that b / ∈ ( ) (non-resonant case). Then the following conditions guarantee the existence of at least three solutions for sufficiently small f: (a) h(x, t, ) -jh/j (x, t, 0) is non-decreasing (resp. non-increasing) in , and sup (x,t, ) jh j (x, t, ) < min{ ∈ ( ); b < }