# American Institute of Mathematical Sciences

February  2020, 25(2): 545-554. doi: 10.3934/dcdsb.2019253

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

 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

Dedicated to Professor Juan José Nieto on the occasion of his 60th birthday.

Received  January 2019 Revised  May 2019 Published  November 2019

Fund Project: 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

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.

Citation: José Ángel Cid, Luís Sanchez. Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 545-554. doi: 10.3934/dcdsb.2019253
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##### References:
Graph of the potential $V$ for the values $r = 12, \, s = 8, \, \alpha = \frac 1 2$ and $\beta = \frac 3 4$
Phase plane for (3) with the values $r = 12, \, s = 8, \, \alpha = \frac 1 2$ and $\beta = \frac 3 4$
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