Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

Lisbeth Carrero; Alexander Quaas

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arXiv · 2012.02313

Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

Lisbeth Carrero·Alexander Quaas

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Abstract

In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in(1/2,1)$, singular nonlinearity, and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we obtain also multiplicity results in a regime where a priori bounds are lost and bifurcation from infinity occurs.

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Lisbeth Carrero, Alexander Quaas. 2020-12-03. Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities. https://arxiv.org/abs/2012.02313

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