Abstract
In this paper we study the existence of periodic solution for a class of beam equation via variational methods. The main difficulty is associated with the fact that the energy functional is strongly indefinite and it is not well defined in the natural space to apply variational methods.
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References
Bartsch, T., Ding, Y.: Periodic solutions of superlinear beam and membrane equations with perturbations from symmetry. Nonlinear Anal. 44, 727–748 (2001)
Bartsch, T., Ding, Y.: On a nonlinear Schrödinger equation with periodic potential. Math. Ann. 313, 15–37 (1999)
Chang, K.C., Sanchez, L.: Nontrivial periodic solutions of a nonlinear beam equation. Math. Methods Appl. Sci. 4, 194–205 (1982)
Cavalcante, M.P., Alves, C.O., Medeiros, E.: A semilinear Schrödinger equation with zero on the boundary of the spectrum and exponential growth in \({\mathbb{R}} ^2\). Commun. Contemp. Math. 21, 1850037 (2019)
Feireisl, E.: Time periodic solutions to a semilinear beam equation. Nonlinear Anal. Theory Methods Appl. 12, 279–290 (1988)
Feireisl, E.: On the existence of infinitely many periodic solutions for an equation of a rectangular thin plate. Czechoslov. Math. J. 37(2), 334–341 (1987)
Holmes, P., Marsden, J.: A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Arch. Ration. Mech. Anal. 76, 135–165 (1981)
Liu, J.Q.: Nonlinear vibration of a beam. Nonlinear Anal. TMA 13, 1139–1148 (1989)
Marsden, J.: Lecture on Geometric Methods in Mathematical Physics, CBMS, vol. 37. SIAM, Philadelphia (1981)
Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005)
Schechter, M.: Nonlinear Schrödinger operators with zero in the spectrum. Z. Angew. Math. Phys. 66, 2125–2141 (2015)
Schechter, M., Zou, W.: Weak linking theorems and Schrödinger equations with critical Sobolev exponent. ESAIM Control Optim. Calc. Var. 9, 601–619 (2003)
Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257, 3802–3822 (2009)
Szulkin, A., Weth, T.: The method of Nehari manifold. In: Gao, D.Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis and Applications, pp. 597–632. International Press, Vienna (2010)
Szilard, R.: Theory and Analysis of Plates. Prentice-Hall, Englewood Cliffs (1974)
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Communicated by Ansgar Jüngel.
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C.O. Alves was partially supported by CNPq/Brazil 304804/2017-7.
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Alves, C.O., de Araújo, B.S.V. & Nóbrega, A.B. Existence of periodic solution for a class of beam equation via variational methods. Monatsh Math 197, 227–256 (2022). https://doi.org/10.1007/s00605-021-01583-z
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DOI: https://doi.org/10.1007/s00605-021-01583-z