On sequences of homoclinic solutions for fractional discrete $ p $-Laplacian equations

Chunming Ju; Giovanni Molica Bisci; Binlin Zhang; Chunming Ju; Giovanni Molica Bisci; Binlin Zhang

doi:10.3934/cam.2023029

Communications in Analysis and Mechanics

2023, Volume 15, Issue 4: 586-597. doi: 10.3934/cam.2023029

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On sequences of homoclinic solutions for fractional discrete $ p $-Laplacian equations

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China
2.
Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Urbino, 61029, Italy
3.
School of Mathematics, Zhejiang Normal University, Jinhua, 321004, P.R. China

Received: 06 September 2023 Revised: 22 September 2023 Accepted: 27 September 2023 Published: 10 October 2023
35R11, 49M25, 35J20

In this paper, we consider the following discrete fractional $ p $-Laplacian equations:

$ \begin{equation*} (-\Delta_{1})^{s}_{p}u(a)+V(a)|u(a)|^{p-2}u(a) = \lambda f(a, u(a)), \; \mbox{in}\ \mathbb{Z}, \end{equation*} $

where $ \lambda $ is the parameter and $ f(a, u(a)) $ satisfies no symmetry assumption. As a result, a specific positive parameter interval is determined by some requirements for the nonlinear term near zero, and then infinitely many homoclinic solutions are obtained by using a special version of Ricceri's variational principle.
- discrete fractional $ p $-Laplacian,
- homoclinic solutions,
- Ricceri's variational principle
Citation: Chunming Ju, Giovanni Molica Bisci, Binlin Zhang. On sequences of homoclinic solutions for fractional discrete $ p $-Laplacian equations[J]. Communications in Analysis and Mechanics, 2023, 15(4): 586-597. doi: 10.3934/cam.2023029

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Abstract

In this paper, we consider the following discrete fractional $ p $-Laplacian equations:

$ \begin{equation*} (-\Delta_{1})^{s}_{p}u(a)+V(a)|u(a)|^{p-2}u(a) = \lambda f(a, u(a)), \; \mbox{in}\ \mathbb{Z}, \end{equation*} $

where $ \lambda $ is the parameter and $ f(a, u(a)) $ satisfies no symmetry assumption. As a result, a specific positive parameter interval is determined by some requirements for the nonlinear term near zero, and then infinitely many homoclinic solutions are obtained by using a special version of Ricceri's variational principle.

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