Tokyo Journal of Mathematics

Application of Local Linking to Asymptotically Linear Wave Equations with Resonance

Shizuo Miyajima and Mieko Tanaka

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Abstract

Existence of a time-periodic solution to a non-linear wave equation with resonance is established by a variational method. We consider the $2\pi$-periodic weak solution to a wave equation $\Box u(x,t)=h(x,t,u(x,t))$ of space dimension 1, where $h(x,t,\xi)$ is asymptotically linear in $\xi$ both as $\xi\to0$ or $\xi\to\infty$, with the co-efficient as $\xi\to\infty$ belonging to $\sigma(\Box)$. It is proved that there are some cases, where the difference of $h(t,x,\xi)$ from its linear approximation is not bounded, that guarantee the existence of a non-trivial weak solutions. The proof is based on local linking theory and $({\it WPS})^*$ condition for the existence of a non-trivial critical point of a functional.

Article information

Source
Tokyo J. Math., Volume 29, Number 1 (2006), 19-43.

Dates
First available in Project Euclid: 20 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1166661865

Digital Object Identifier
doi:10.3836/tjm/1166661865

Mathematical Reviews number (MathSciNet)
MR2258270

Zentralblatt MATH identifier
1109.58020

Subjects
Primary: 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)
Secondary: 35L05: Wave equation 35L35: Initial-boundary value problems for higher-order hyperbolic equations 47J30: Variational methods [See also 58Exx]

Citation

Miyajima, Shizuo; Tanaka, Mieko. Application of Local Linking to Asymptotically Linear Wave Equations with Resonance. Tokyo J. Math. 29 (2006), no. 1, 19--43. doi:10.3836/tjm/1166661865. https://projecteuclid.org/euclid.tjm/1166661865


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