Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach

José Ángel Cid; Luís Sanchez

doi:10.3934/dcdsb.2019253

Article Contents

2020, Volume 25, Issue 2: 545-554. Doi: 10.3934/dcdsb.2019253

This issue Previous Article A Favard type theorem for Hurwitz polynomials Next Article Existence of homoclinic solutions for a nonlinear fourth order

$ p $

-Laplacian difference equation

Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach

José Ángel Cid^1, , and
Luís Sanchez^2,

1.
Departamento de Matemáticas, Universidade de Vigo, 32004, Campus de Ourense, Spain
2.
CMAFcIO - Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

^* Corresponding author: José Ángel Cid
^* Corresponding author: José Ángel Cid

Received: December 31, 2018

Revised: April 30, 2019

Early access: November 2019

Published: February 2020

J. Á. Cid was partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Project MTM2017-85054-C2-1-P and L. Sanchez was supported by Fundação para a Ciência e a Tecnologia, UID/MAT/04561/2019

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Abstract

We deal with the existence of nonnegative and nontrivial $ T $–periodic solutions for the equation $ x'' = r(t)x^{\alpha}-s(t)x^{\beta} $ where $ r $ and $ s $ are continuous $ T $–periodic functions and $ 0<\alpha<\beta<1 $. This equation has been studied in connection with the valveless pumping phenomenon and we will take advantage of its variational structure in order to guarantee its solvability by means of the mountain pass theorem of Ambrosetti and Rabinowitz.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 34B18.

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Figure 1. Graph of the potential $ V $ for the values $ r = 12, \, s = 8, \, \alpha = \frac 1 2 $ and $ \beta = \frac 3 4 $

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Figure 2. Phase plane for (3) with the values $ r = 12, \, s = 8, \, \alpha = \frac 1 2 $ and $ \beta = \frac 3 4 $

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