p-Laplacian Equations on Locally Finite Graphs

• Published in: Acta mathematica Sinica. English series Vol. 37; no. 11; pp. 1645 - 1678
• Main Authors: Han, Xiao Li, Shao, Meng Qiu
• Format: Journal Article
• Language: English
• Published: Beijing Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.11.2021; Springer Nature B.V
• Subjects: Dirichlet problem; Graphs; Laplace equation; Mathematics; Mathematics and Statistics; Mountains; 35J20; locally finite graph; 35J05; ground state solution; 35J60; Laplacian equation
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-021-9523-5

This paper is mainly concerned with the following nonlinear p -Laplacian equation − Δ p ( x ) + ( λ a ( x ) + 1 ) ∣ u ∣ p − 2 ( x ) u ( x ) = f ( x , u ( x ) ) , i n 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 { − Δ p ( x ) + ∣ u ∣ p − 2 ( x ) u ( x ) = f ( x , u ( x ) ) , i n Ω u ( x ) = 0 o n ∂ Ω where Ω = { x ∈ V : a ( x ) = 0} is the potential well and ∂ Ω denotes the the boundary of Ω.