Standing Waves of the NLS Equation on Locally Finite Weighted Graphs: A Linking Geometry Approach

Abstract

We investigate wave solutions of the nonlinear Schrödinger equation (H-λ)u=f(x,u) on a locally finite weighted graph, where H is the associated Schrödinger operator. The central hypothesis is a tail summability condition 1/V∈l_m^1 (V) on the effective potential V, which, together with canonical compactifiability of the graph, yields a compact Sobolev-type embedding E↪lᵖₘ(V) for the full range 1≤p0 and is strictly incomparable with the measure-theoretic potential conditions in the existing literature, thereby covering a genuinely broader class of weighted graphs. Under the Ambrosetti–Rabinowitz superlinearity condition on f, we establish the Palais–Smale condition for the associated energy functional for every λ∈R and verify the linking geometry through a spectral decomposition of the energy space. An application of the Linking theorem then yields at least one nontrivial solution for every value of the spectral parameter. When the nonlinearity is additionally odd, the Symmetric Mountain Pass Theorem produces infinitely many pairs of solutions.