Homoclinic solutions of discrete $ \phi $-Laplacian equations with mixed nonlinearities

Genghong Lin; Zhan Zhou

doi:10.3934/cpaa.2018082

Article Contents

2018, Volume 17, Issue 5: 1723-1747. Doi: 10.3934/cpaa.2018082

This issue Previous Article Next Article On existence and nonexistence of positive solutions of an elliptic system with coupled terms

Homoclinic solutions of discrete $ \phi $-Laplacian equations with mixed nonlinearities

Genghong Lin and
Zhan Zhou^,

School of Mathematics and Information Science, Guangzhou University, Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China

* Corresponding author
* Corresponding author

Received: March 2017

Revised: November 2017

Published: April 2018

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Abstract

By using critical point theory, we obtain some new sufficient conditions on the existence of homoclinic solutions of a class of nonlinear discrete $ \phi $-Laplacian equations with mixed nonlinearities for the potentials being periodic or being unbounded, respectively. And we prove it is also necessary in some special cases. In addition, multiplicity results of homoclinic solutions for nonlinear discrete $ \phi $-Laplacian equations with unbounded potentials have also been considered. In our paper, the nonlinearities can be mixed super $ p $-linear with asymptotically $ p $-linear at $ ∞ $ for $ p≥ 1 $. To the best of our knowledge, there is no such result for the existence of homoclinic solutions with discrete $ \phi $-Laplacian before. Finally, an extension has also been considered.

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Mathematics Subject Classification: Primary: 39A14; Secondary: 34C37.

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