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Fisher-Kolmogorov type perturbations of the relativistic operator: differential vs. difference


Authors: Petru Jebelean and Călin Şerban
Journal: Proc. Amer. Math. Soc. 146 (2018), 2005-2014
MSC (2010): Primary 34B15, 34C25, 39A10, 39A23
DOI: https://doi.org/10.1090/proc/13978
Published electronically: January 26, 2018
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Abstract: We are concerned with the existence of multiple periodic solutions for differential equations involving Fisher-Kolmogorov perturbations of the relativistic operator of the form

$\displaystyle -\left [\phi (u')\right ]'=\lambda u(1-\vert u\vert^q),$    

as well as for difference equations, of type

$\displaystyle -\Delta \left [\phi (\Delta u(n-1))\right ]=\lambda u(n)(1-\vert u(n)\vert^q);$    

here $ q>0$ is fixed, $ \Delta $ is the forward difference operator, $ \lambda >0$ is a real parameter and

$\displaystyle \displaystyle \phi (y)=\frac {y}{\sqrt {1- y^2}}\quad (y\in (-1,1)).$    

The approach is variational and relies on critical point theory for convex, lower semicontinuous perturbations of $ C^1$-functionals.

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Additional Information

Petru Jebelean
Affiliation: Department of Mathematics, West University of Timişoara, 4, Boulevard, V. Pârvan 300223 - Timişoara, Romania
Email: petru.jebelean@e-uvt.ro

Călin Şerban
Affiliation: Department of Mathematics, West University of Timişoara, 4, Boulevard, V. Pârvan 300223 - Timişoara, Romania
Email: cserban2005@yahoo.com

DOI: https://doi.org/10.1090/proc/13978
Keywords: Relativistic operator, Fisher-Kolmogorov nonlinearities, difference equations, periodic solution, critical point, Palais-Smale condition, Krasnoselskii genus
Received by editor(s): June 24, 2017
Published electronically: January 26, 2018
Dedicated: Dedicated to Jean Mawhin for his 75th anniversary
Communicated by: Joachim Krieger
Article copyright: © Copyright 2018 American Mathematical Society

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