HOMOCLINIC SOLUTIONS OF NONPERIODIC FOURTH-ORDER $\phi_{c}$-LAPLACIAN PARTIAL DIFFERENCE EQUATIONS

Sha Li; Yuhua Long

doi:10.11948/20260134

2027 Volume 17 Issue 2

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Article navigation > Journal of Applied Analysis & Computation > 2027 > 17(2): 868-883

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Sha Li, Yuhua Long. HOMOCLINIC SOLUTIONS OF NONPERIODIC FOURTH-ORDER $\phi_{c}$-LAPLACIAN PARTIAL DIFFERENCE EQUATIONS[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 868-883. doi: 10.11948/20260134

Citation:

Sha Li, Yuhua Long. HOMOCLINIC SOLUTIONS OF NONPERIODIC FOURTH-ORDER $\phi_{c}$-LAPLACIAN PARTIAL DIFFERENCE EQUATIONS[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 868-883. doi: 10.11948/20260134

HOMOCLINIC SOLUTIONS OF NONPERIODIC FOURTH-ORDER $\phi_{c}$-LAPLACIAN PARTIAL DIFFERENCE EQUATIONS

Sha Li^1,2,,
Yuhua Long^1,2, ,

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2.
Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China

Author Bio: Email: LSha@e.gzhu.edu.cn(S. Li)
Corresponding author: Email: sxlongyuhua@gzhu.edu.cn(Y. Long)
Fund Project: The authors were supported by National Natural Science Foundation of China (12471177)

Abstract

Abstract

In this paper, we investigate the existence of nontrivial homoclinic solutions for a class of nonperiodic fourth-order partial difference equations involving $\phi_{c}$-Laplacian. By the Mountain Pass Lemma and Fountain Theorem, we establish two key results: (1) Under mild nonlinear growth conditions, the equation admits at least one nontrivial homoclinic solution; (2) for odd symmetric nonlinear terms, the equation possesses infinitely many homoclinic solutions. A critical innovation lies in relaxing the classical Ambrosetti-Rabinowitz (AR) condition to a more general form, enabling the applications of our results to nonlinearities (e.g., logarithmic terms) that fail the AR condition. Additionally, the proposed framework is extendable to higher-order $\phi_{c}$-Laplacian partial difference equations. Two concrete examples are provided to verify the theoretical validity and practical applicability of our theorems.
- Nontrivial homoclinic solution /
- ϕ_c-Laplacian /
- fourth-order partial difference equation /
- Mountain Pass Lemma /
- Fountain Theorem
MSC: 39A14, 34C37

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References

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