P-Laplacian Equations on Locally Finite Graphs

Abstract

This paper is mainly concerned with the following nonlinear p-Laplacian equation $$ - {\Delta _p}u(x) + (\lambda a(x) + 1){\left| u \right|^{p - 2}}(x)u(x) = f(x,u(x)),\;\;\;{\rm{in}}\;V$$ 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 $$\left\{ {\matrix{ { - {\Delta _p}u(x) + {{\left| u \right|}^{p - 2}}(x)u(x) = f(x,u(x)),} \;\;\;\;\;\;\;\; {{\rm{in}}\;\Omega ,} \hfill \cr {u(x) = 0,} \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\; {{\rm{on}}\;\partial \Omega ,} \hfill \cr } } \right.$$ where Ω = {x ∈ V: a(x) = 0} is the potential well and ∂Ω denotes the the boundary of Ω.