Convergence of least energy sign-changing solutions for logarithmic Schrödinger equations on locally finite graphs

Abstract

In this paper, we study the following logarithmic Schrödinger equation −Δu+λa(x)u=ulogu2inVon a connected locally finite graph G=(V,E), where Δ denotes the graph Laplacian, λ>0 is a constant, and a(x)≥0 represents the potential. Using variational techniques in combination with the Nehari manifold method based on directional derivative, we can prove that, there exists a constant λ0>0 such that for all λ≥λ0, the above problem admits a least energy sign-changing solution uλ. Moreover, as λ→+∞, we prove that the solution uλ converges to a least energy sign-changing solution of the following Dirichlet problem −Δu=ulogu2inΩ,u(x)=0on∂Ω,where Ω={x∈V:a(x)=0} is the potential well.