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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Infinitely many periodic solutions for the equation: $ u\sb {tt}-u\sb {xx}\pm \vert u\vert \sp {p-1}u=f(x,t)$. II


Author: Kazunaga Tanaka
Journal: Trans. Amer. Math. Soc. 307 (1988), 615-645
MSC: Primary 35B10; Secondary 35L70, 58E05, 58G16
MathSciNet review: 940220
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Abstract: Existence of forced vibrations of nonlinear wave equation:

\begin{displaymath}\begin{array}{*{20}{c}} {{u_{tt}} - {u_{xx}} \pm \vert u{\ver... ...,t) \in (0,\,\pi ) \times {\mathbf{R}},} \hfill \\ \end{array} \end{displaymath}

is considered. For all $ p \in (1,\,\infty )$ and $ f(x,\,t) \in {L^{(p + 1)/p}}$, existence of infinitely many periodic solutions is proved. This improves the results of the author [29, 30].

We use variational methods to show the existence result. Minimax arguments and energy estimates for the corresponding functional play an essential role in the proof.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0940220-X
PII: S 0002-9947(1988)0940220-X
Keywords: Nonlinear wave equation, periodic solution, variational method, minimax method, perturbation
Article copyright: © Copyright 1988 American Mathematical Society