Multiplicity of positive solutions for a superlinear-singular (p,q)-Laplacian equation on locally finite graphs

We investigate the multiplicity of positive solutions for the superlinear singular [Formula: see text]-Laplacian equation [Formula: see text] on a weighted locally finite graph [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] is a parameter. The potential functions [Formula: see text] and [Formula: see text] satisfy some suitable conditions. The function [Formula: see text], [Formula: see text]. Besides, the nonlinear term [Formula: see text] exhibits a more general superlinear growth condition that is weaker than the Ambrosetti–Rabinowitz type condition. We establish the existence of two distinct solutions through variational methods combined with perturbation techniques. More precisely, the first solution is obtained via a local minimization argument. Furthermore, the existence of the second solution is established by employing the Mountain Pass Lemma. We overcome some difficulties which are caused by: [Formula: see text] the singular term; [Formula: see text] the lack of compactness of Sobolev embedding; [Formula: see text] the weakened Ambrosetti–Rabinowitz condition.