Ground states for logarithmic Schr\"{o}dinger equations on locally finite graphs

Abstract

In this paper, we study the following logarithmic Schr\"{o}dinger equation \[ -\Delta u+a(x)u=u\log u^2\ \ \ \ \mbox{in }V, \] where $\Delta$ is the graph Laplacian, $G=(V,E)$ is a connected locally finite graph, the potential $a: V\to \mathbb{R}$ is bounded from below and may change sign. We first establish two Sobolev compact embedding theorems in the case when different assumptions are imposed on $a(x)$. It leads to two kinds of associated energy functionals, one of which is not well-defined under the logarithmic nonlinearity, while the other is $C^1$. The existence of ground state solutions are then obtained by using the Nehari manifold method and the mountain pass theorem respectively.