Abstract
We generalize two embedding theorems and investigate the existence and multiplicity of nontrivial solutions for a $(p,q)$-Laplacian coupled system with perturbations and two parameters $\lambda_{1}$ and $\lambda_{2}$ on locally finite graph. By using the Ekeland's variational principle, we obtain that system has at least one nontrivial solution when the nonlinear term satisfies the sub-$(p,q)$ conditions. We also obtain a necessary condition for the existence of semi-trivial solutions to the system. Moreover, by using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one solution of positive energy and one solution of negative energy when the nonlinear term satisfies the super-$(p,q)$ conditions which is weaker than the well-known Ambrosetti–Rabinowitz condition. Especially, in all of the results, we present the concrete ranges of the parameters $\lambda_{1}$ and $\lambda_{2}$.
Funding Statement
This project is supported by Yunnan Fundamental Research Projects in China (grant No. 202301AT070465) and Xingdian Talent Support Program for Young Talents of Yunnan Province in China.
Citation
Ping Yang. Xingyong Zhang. "Existence and Multiplicity of Nontrivial Solutions for a $(p,q)$-Laplacian System on Locally Finite Graphs." Taiwanese J. Math. 28 (3) 551 - 588, June, 2024. https://doi.org/10.11650/tjm/240201
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