Convergence of least energy sign-changing solutions for logarithmic Schr\"{o}dinger equations on locally finite graphs

Abstract

In this paper, we study the following logarithmic Schr\"{o}dinger equation \[ -\Delta u+\lambda a(x)u=u\log u^2\ \ \ \ \mbox{ in }V \] on a connected locally finite graph $G=(V,E)$, where $\Delta$ denotes the graph Laplacian, $\lambda > 0$ 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 $\lambda_0>0$ such that for all $\lambda\geq\lambda_0$, the above problem admits a least energy sign-changing solution $u_{\lambda}$. Moreover, as $\lambda\to+\infty$, we prove that the solution $u_{\lambda}$ converges to a least energy sign-changing solution of the following Dirichlet problem \[\begin{cases} -\Delta u=u\log u^2~~~&\mbox{ in }\Omega,\\ u(x)=0~~~&\mbox{ on }\partial\Omega, \end{cases}\] where $\Omega=\{x\in V: a(x)=0\}$ is the potential well.