Ground States for Logarithmic Schrödinger Equations on Locally Finite Graphs.

In this paper, we study the following logarithmic Schrödinger equation: - Δ u + a (x) u = u log u 2 in V , where Δ is the graph Laplacian, G = (V , E) is a connected locally finite graph, the potential a : V → 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). This 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. [ABSTRACT FROM AUTHOR]