Abstract
This paper is mainly concerned with the following nonlinear p-Laplacian equation
on a locally finite graph G = (V, E) with more general nonlinear term, where Δp is the discrete p-Laplacian on graphs, p ≥ 2. Under some suitable conditions on f and a(x), we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλ via the method of Nehari manifold, for any λ > 1. In addition, as λ → +∞, we prove that the solution uλ converge to a solution of the following Dirichlet problem
where Ω = {x ∈ V:a(x) = 0} is the potential well and ∂Ω denotes the the boundary of Ω.
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Acknowledgements
The authors are grateful to Prof. Zhao Liang for his encouragement and valuable suggestions. The authors would also like to thank the referees for a thoughtful reading.
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Han, X.L., Shao, M.Q. p-Laplacian Equations on Locally Finite Graphs. Acta. Math. Sin.-English Ser. 37, 1645–1678 (2021). https://doi.org/10.1007/s10114-021-9523-5
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DOI: https://doi.org/10.1007/s10114-021-9523-5