Existence and convergence of solutions to p-Laplace equations on locally finite graphs

2023/06/25 by Shao, Mengqiu, Yang, Yunyan, Zhao, Liang
#35A15 #35Q55 #35R02 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2306.14121

Abstract

We are mainly concerned with the nonlinear p-Laplace equation -Δ_pu+ρ|u|^p-2u=ψ(x,u) on a locally finite graph G=(V,E), where p belongs to (1, +∞). We obtain existence of positive solutions and positive ground state solutions by using the mountain-pass theorem and the Nehari manifold respectively. Moreover, we also analyze the asymptotic behavior for a sequence of positive ground state solutions. Compared with all the existing relevant works, our results have made essential improvements in at least three aspects: (i) p can take any value in (1, +∞); (ii) the conditions on the graph G and the potential ρ are relaxed; (iii) for the existence of positive solutions, the growth condition in previous works on the nonlinear term ψ(x,s) as s→ +∞ is removed.

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