Existence of nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearities on locally finite graphs

Ping Yang; Xingyong Zhang; Ping Yang; Xingyong Zhang

doi:10.3934/era.2023377

Electronic Research Archive

2023, Volume 31, Issue 12: 7473-7495. doi: 10.3934/era.2023377

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Existence of nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearities on locally finite graphs

Ping Yang ¹,
Xingyong Zhang ^{1,2
,
,}

1.
Faculty of Science, Kunming University of Science and Technology, Kunming 650500, Yunnan, China
2.
Research Center for Mathematics and Interdisciplinary Sciences, Kunming University of Science and Technology, Kunming 650500, Yunnan, China

Received: 16 August 2023 Revised: 24 October 2023 Accepted: 27 October 2023 Published: 23 November 2023

We discuss a poly-Laplacian system involving concave-convex nonlinearities and parameters subject to the Dirichlet boundary condition on locally finite graphs. It is obtained that the system admits at least one nontrivial solution of positive energy and one nontrivial solution of negative energy based on the mountain pass theorem and the Ekeland's variational principle. We also obtain an estimate about semi-trivial solutions. Moreover, by using a result due to Brown et al., which is based on the fibering method and the Nehari manifold, we get the existence of the ground-state solution to the single equation corresponding to the poly-Laplacian system. Especially, we present some ranges of parameters for all of the results.
- poly-Laplacian system,
- concave-convex nonlinearities,
- mountain pass theorem,
- Ekeland's variational principle,
- locally finite graph
Citation: Ping Yang, Xingyong Zhang. Existence of nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearities on locally finite graphs[J]. Electronic Research Archive, 2023, 31(12): 7473-7495. doi: 10.3934/era.2023377

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Abstract

We discuss a poly-Laplacian system involving concave-convex nonlinearities and parameters subject to the Dirichlet boundary condition on locally finite graphs. It is obtained that the system admits at least one nontrivial solution of positive energy and one nontrivial solution of negative energy based on the mountain pass theorem and the Ekeland's variational principle. We also obtain an estimate about semi-trivial solutions. Moreover, by using a result due to Brown et al., which is based on the fibering method and the Nehari manifold, we get the existence of the ground-state solution to the single equation corresponding to the poly-Laplacian system. Especially, we present some ranges of parameters for all of the results.

References

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