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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 solutions to fourth order superlinear periodic problems


Authors: Monica Conti, Susanna Terracini and Gianmaria Verzini
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3283-3300
MSC (2000): Primary 34B15; Secondary 58E05, 47J10
Published electronically: December 12, 2003
MathSciNet review: 2052950
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Abstract: We present a new min-max approach to the search of multiple $T$-periodic solutions to a class of fourth order equations

\begin{displaymath}u^{iv}(t)-c u''(t)=f(t,u(t)),\hspace{5mm}t\in[0,T],\end{displaymath}

where $f(t,u)$ is continuous, $T$-periodic in $t$ and satisfies a superlinearity assumption when $\vert u\vert\to\infty$. For every $n\in\mathbb{N}$, we prove the existence of a $T$-periodic solution having exactly $2n$ zeroes in $(0,T]$.


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Additional Information

Monica Conti
Affiliation: Dipartimento di Matematica del Politecnico, piazza Leonardo da Vinci, 32 - 20133 Milano (I), Italy
Email: monica.conti@polimi.it

Susanna Terracini
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano (I), Italy
Email: suster@matapp.unimib.it

Gianmaria Verzini
Affiliation: Dipartimento di Matematica del Politecnico, piazza Leonardo da Vinci, 32 - 20133 Milano (I), Italy
Email: gianmaria.verzini@polimi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03514-1
PII: S 0002-9947(03)03514-1
Keywords: Oscillating solutions, fourth order equations, boundary value problems, variational methods
Received by editor(s): May 25, 2001
Received by editor(s) in revised form: March 21, 2003
Published electronically: December 12, 2003
Additional Notes: This research was supported by MURST project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”
Article copyright: © Copyright 2003 American Mathematical Society