Periodic solutions of p-Laplacian equations via rotation numbers

Shuang Wang; Dingbian Qian

doi:10.3934/cpaa.2021060

Article Contents

2021, Volume 20, Issue 5: 2117-2138. Doi: 10.3934/cpaa.2021060

This issue Previous Article A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties Next Article

Periodic solutions of p-Laplacian equations via rotation numbers

Shuang Wang^1, , and
Dingbian Qian^2,

1.
School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224051, China
2.
School of Mathematical Sciences, Soochow University, Suzhou 215006, China

^* Corresponding author
^* Corresponding author

Received: February 29, 2020

Revised: January 31, 2021

Early access: April 2021

Published: May 2021

This work is supported by the National Natural Science Foundation of China (No. 11901507, No. 12071327 and No. 11671287), the Natural Science Foundation of Jiangsu Province (No. BK20181058), and the Qing Lan Project of the Jiangsu Higher Education Institutions of China

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Abstract

We investigate the existence and multiplicity of periodic solutions of the $ p $-Laplacian equation $ \left(\phi_p(x')\right)'+f(t, x) = 0 $. Both asymptotically linear and partially superlinear nonlinearities are studied, in absence of global existence and uniqueness conditions on the solutions of the associated Cauchy problems and the sign assumption on $ f $. We use a approach of rotation number in the $ p $-polar coordinates transformation, together with the phase-plane analysis of the rotational properties of large solutions and a recent version of Poincaré-Birkhoff theorem for Hamiltonian systems, for obtaining multiplicity results of $ p $-Laplacian equation in terms of the gap between the rotation numbers of referred piecewise $ p $-linear systems at zero and infinity.

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Mathematics Subject Classification: Primary: 34C25, 34B15; Secondary: 34D15.

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Figure 1. The trajectories in regions $ \mathcal{D}_{1} $ and $ \mathcal{D}_{2} $

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Figure 2. Trajectory intersects $ y = 0 $ and $ y = -\delta $

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Figure 3. Trajectory intersects $ y = 0 $ and $ y = \delta $

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