Nontrivial solutions for a generalized poly-Laplacian system on finite graphs

• Published in: Demonstratio mathematica Vol. 58; no. 1; pp. 309 - 323
• Main Authors: Qi, Wanting, Zhang, Xingyong
• Format: Journal Article
• Language: English
• Published: De Gruyter 31.07.2025
• Subjects: 35B38; 35J50; 35J62; 35R02; Clark’s theorem; coupled system on finite graphs; existence; infinitely many nontrivial solutions; local nonlinearity near origin; Mountain pass lemma; poly-Laplacian
• ISSN: 2391-4661; 2391-4661
• DOI: 10.1515/dema-2025-0148

We investigate the existence and multiplicity of solutions for a class of the generalized coupled system involving poly-Laplacian and the parameter on finite graphs. By using the Mountain pass lemma together with the cut-off technique, we obtain that system has at least a nontrivial weak solution for every large parameter when the nonlinear term satisfies superlinear growth conditions only in a neighborhood of origin point (0, 0). We also obtain a concrete form for the lower bound of and the trend of with the change of . Moreover, by using a revised Clark’s theorem together with cut-off technique, we obtain that system has a sequence of solutions tending to 0 for every when the nonlinear term satisfies sublinear growth conditions only in a neighborhood of origin point (0, 0).