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 \[ -Δu+λa(x)u=u\log u^2\ \ \ \ \mbox{ in }V \] on a connected locally finite graph $G=(V,E)$, where $Δ$ denotes the graph Laplacian, $λ> 0$ is a constant, and $a(x) \geq 0$ 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 $λ_0>0$ such that for all $λ\geqλ_0$, the above problem admits a least energy sign-changing solution $u_λ$. Moreover, as $λ\to+\infty$, we prove that the solution $u_λ$ converges to a least energy sign-changing solution of the following Dirichlet problem \[\begin{cases} -Δu=u\log u^2~~~&\mbox{ in }Ω,\\ u(x)=0~~~&\mbox{ on }\partialΩ, \end{cases}\] where $Ω=\{x\in V: a(x)=0\}$ is the potential well.