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 $\hat{I}”$ denotes the graph Laplacian, $\hat{I}»>0$ is a constant, and $a(x)\ge 0$ 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 $\hat{I}{»}_0>0$ such that for all $\hat{I}»\ge \hat{I}{»}_0$, the above problem admits a least energy sign-changing solution $u_{\hat{I}}»$. Moreover, as $\hat{I}»\to +\infty$, we prove that the solution $u_{\hat{I}}»$ 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 $\hat{I}\text{c◯}=\{x\in V:a(x)=0\}$ is the potential well.

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