On the structure of periodic eigenvalues of the vectorial \(p\)-Laplacian. (English) Zbl 1505.37071

Liu, Changjian; Zhang, Meirong

Nonlinearity 35, No. 5, 2206-2240 (2022).

Authors’ abstract: In this paper we will solve an open problem raised by R. Manásevich and J. Mawhin [Adv. Differ. Equ. 5, No. 10–12, 1289–1318 (2000; Zbl 0992.34063)] twenty years ago on the structure of the periodic eigenvalues of the vectorial \(p\)-Laplacian. This is an Euler-Lagrangian equation on the plane or in higher dimensional Euclidean spaces. The main result obtained is that for any exponent p other than 2, the vectorial \(p\)-Laplacian on the plane will admit infinitely many different sequences of periodic eigenvalues with a given period. These sequences of eigenvalues are constructed using the notion of scaling momenta we will introduce. The whole proof is based on the complete integrability of the equivalent Hamiltonian system, the tricky reduction to two-dimensional dynamical systems, and a number-theoretical distinguishing between different sequences of eigenvalues. Some numerical simulations to the new sequences of eigenvalues and eigenfunctions will be given. Several further conjectures towards to the panorama of the spectral sets will be imposed.

Reviewer: Abderrazek Benhassine (Monastir)

MSC:

37J46	Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70H03	Lagrange’s equations
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B38	Critical points of functionals in context of PDEs (e.g., energy functionals)
70H06	Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
34C14	Symmetries, invariants of ordinary differential equations
34L40	Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B15	Nonlinear boundary value problems for ordinary differential equations
70G60	Dynamical systems methods for problems in mechanics

Keywords:

vectorial \(p\)-Laplacian; periodic eigenvalues; eigenfunctions; Hamiltonian systems with two degrees of freedom; complete integrability; scaling angular momenta; reduced dynamical systems

Citations:

Zbl 0992.34063

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