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Nonnegative oscillations for a class of differential equations without uniqueness: a variational approach. (English) Zbl 1433.34060

Summary: 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.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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