Global existence and multiplicity of solutions for logarithmic Schrödinger equations on graphs

We consider the following logarithmic Schrödinger equation $$-\mathrm{\Delta }u+h(x)u=u\log u^2$$ on a locally finite graph $G=(V,E)$, where $\mathrm{\Delta }$ is a discrete Laplacian operator on the graph, $h$ is the potential function. Different from the classical methods in Euclidean space, we obtain the existence of global solutions to the equation by using the variational method from local to global, which is inspired by the works of Lin and Yang in \cite{LinYang}. In addition, when the potential function $h$ is sign-changing, we prove that the equation admits infinitely many solutions with high energy by using the symmetric mountain pass theorem. We extend the classical results in Euclidean space to discrete graphs.