This paper is mainly concerned with the following nonlinear p-Laplacian equation $$ - {\Delta _p}u(x) + (\lambda a(x) + 1){\left| u \right|^{p - 2}}(x)u(x) = f(x,u(x)),\;\;\;{\rm{in}}\;V$$ − Δ p u ( x ) + ( λ a ( x ) + 1 ) | u | p − 2 ( x ) u ( x ) = f ( x , u ( x ) ) , in V on a locally finite graph G =( V, E ) with more general nonlinear term, where Δ p 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 u λ via the method of Nehari manifold, for any λ > 1. In addition, as λ → + ∞, we prove that the solution u λ converge to a solution of the following Dirichlet problem $$\left\{ {\matrix{ { - {\Delta _p}u(x) + {{\left| u \right|}^{p - 2}}(x)u(x) = f(x,u(x)),} \;\;\;\;\;\;\;\; {{\rm{in}}\;\Omega ,} \hfill \cr {u(x) = 0,} \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\; {{\rm{on}}\;\partial \Omega ,} \hfill \cr } } \right.$$ { − Δ p u ( x ) + | u | p − 2 ( x ) u ( x ) = f ( x , u ( x ) ) , in Ω , u ( x ) = 0 , on ∂ Ω , where Ω = { x ∈ V: a(x) = 0} is the potential well and ∂ Ω denotes the the boundary of Ω .

中文翻译：

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局部有限图上的 p-拉普拉斯方程

本文主要关注以下非线性p-Laplacian方程$$ - {\Delta _p}u(x) + (\lambda a(x) + 1){\left| u \right|^{p - 2}}(x)u(x) = f(x,u(x)),\;\;\;{\rm{in}}\;V$$ − Δ pu ( x ) + ( λ a ( x ) + 1 ) | 你| p − 2 ( x ) u ( x ) = f ( x , u ( x ) ) ，在具有更一般非线性项的局部有限图 G =( V, E ) 上的 V 中，其中 Δ p 是离散 p-拉普拉斯算子在图上，p ≥ 2。在 f 和 a ( x ) 上的一些合适条件下，我们可以证明该方程允许通过山口定理的正解和通过 Nehari 流形方法的基态解 u λ，对于任何λ > 1。此外，当 λ → + ∞ 时，我们证明解 u λ 收敛到以下 Dirichlet 问题 $$\left\{ {\matrix{ { - {\Delta _p}u(x) + {{\左| u \right|}^{p - 2}}(x)u(x) = f(x,u(x)),} \;\;\;\;\;\;\;\; {{\rm{in}}\;\Omega ,} \hfill \cr {u(x) = 0,} \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\; {{\rm{on}}\;\partial \Omega ,} \hfill \cr } } \right.$$ { − Δ pu ( x ) + | 你| p − 2 ( x ) u ( x ) = f ( x , u ( x ) ) , in Ω , u ( x ) = 0 , on ∂ Ω , 其中 Ω = { x ∈ V: a(x) = 0}是势阱，∂ Ω 表示Ω 的边界。