Infinitely many solutions to fourth order superlinear periodic problems

Conti, Monica; Terracini, Susanna; Verzini, Gianmaria

doi:10.1090/S0002-9947-03-03514-1

Infinitely many solutions to fourth order superlinear periodic problems
HTML articles powered by AMS MathViewer

by Monica Conti, Susanna Terracini and Gianmaria Verzini

Trans. Amer. Math. Soc. 356 (2004), 3283-3300

DOI: https://doi.org/10.1090/S0002-9947-03-03514-1

Published electronically: December 12, 2003

PDF | Request permission

Abstract:

We present a new min–max approach to the search of multiple $T$–periodic solutions to a class of fourth order equations \[ u^{iv}(t)-c u''(t)=f(t,u(t)),\hspace {5mm}t\in [0,T],\] where $f(t,u)$ is continuous, $T$–periodic in $t$ and satisfies a superlinearity assumption when $|u|\to \infty$. For every $n\in \mathbb {N}$, we prove the existence of a $T$–periodic solution having exactly $2n$ zeroes in $(0,T]$.

References

A.U. Afuwape, J. Mawhin, F. Zanolin, An existence theorem for periodic solutions and applications to some nonlinear ordinary and delay equations, preprint
Thomas Bartsch and Michel Willem, Infinitely many radial solutions of a semilinear elliptic problem on $\textbf {R}^N$, Arch. Rational Mech. Anal. 124 (1993), no. 3, 261–276. MR 1237913, DOI 10.1007/BF00953069
D. Bonheure, L. Sanchez, M. Tarallo, S. Terracini, Heteroclinics connections between nonconsecutive equilibria of a fourth order differential equation, Calculus of Variation and PDE’s, to appear
Julia Chaparova, Existence and numerical approximations of periodic solutions of semilinear fourth-order differential equations, J. Math. Anal. Appl. 273 (2002), no. 1, 121–136. MR 1933020, DOI 10.1016/S0022-247X(02)00216-0
Monica Conti, Luca Merizzi, and Susanna Terracini, Radial solutions of superlinear equations on $\textbf {R}^N$. I. A global variational approach, Arch. Ration. Mech. Anal. 153 (2000), no. 4, 291–316. MR 1773818, DOI 10.1007/s002050050015
Monica Conti and Susanna Terracini, Radial solutions of superlinear equations on $\textbf {R}^N$. II. The forced case, Arch. Ration. Mech. Anal. 153 (2000), no. 4, 317–339. MR 1773819, DOI 10.1007/s002050050016
Monica Conti, Susanna Terracini, and Gianmaria Verzini, Nodal solutions to a class of nonstandard superlinear equations on $\Bbb R^N$, Adv. Differential Equations 7 (2002), no. 3, 297–318. MR 1867689
Felix L. Garcia, Periodic solutions of 4th order O.D.E, Dynam. Contin. Discrete Impuls. Systems 7 (2000), no. 2, 239–254. MR 1744968
P. Habets, L. Sanchez, M. Tarallo, S. Terracini, Heteroclinics for a class of fourth order conservative differential equations, Equadiff 10, Prague 2001, CD-ROM Proceedings
A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev. 32 (1990), no. 4, 537–578. MR 1084570, DOI 10.1137/1032120
J. Mawhin and F. Zanolin, A continuation approach to fourth order superlinear periodic boundary value problems, Topol. Methods Nonlinear Anal. 2 (1993), no. 1, 55–74. MR 1245478, DOI 10.12775/TMNA.1993.030
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
V. J. Mizel, L. A. Peletier, and W. C. Troy, Periodic phases in second-order materials, Arch. Ration. Mech. Anal. 145 (1998), no. 4, 343–382. MR 1664530, DOI 10.1007/s002050050133
Zeev Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math. 105 (1961), 141–175. MR 123775, DOI 10.1007/BF02559588
L. A. Peletier and W. C. Troy, Spatial patterns described by the extended Fisher-Kolmogorov (EFK) equation: kinks, Differential Integral Equations 8 (1995), no. 6, 1279–1304. MR 1329841
L. A. Peletier and W. C. Troy, A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation, Topol. Methods Nonlinear Anal. 6 (1995), no. 2, 331–355. MR 1399544, DOI 10.12775/TMNA.1995.049
L. A. Peletier and W. C. Troy, Spatial patterns described by the extended Fisher-Kolmogorov equation: periodic solutions, SIAM J. Math. Anal. 28 (1997), no. 6, 1317–1353. MR 1474217, DOI 10.1137/S0036141095280955
L. A. Peletier and W. C. Troy, Multibump periodic travelling waves in suspension bridges, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 3, 631–659. MR 1632819, DOI 10.1017/S0308210500021661
B.R. Rynne, Global bifurcation for $2m$-th order boundary value problems and infinitely many solutions of superlinear problems, preprint
D. Smets, J.B. van den Berg, Homoclinic solutions for Swift–Hohenberg and suspension bridge type equations, J. Diff. Eq. 184 (2002), 78–96
Michael Struwe, Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math. (Basel) 39 (1982), no. 3, 233–240. MR 682450, DOI 10.1007/BF01899529