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

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.