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

Source abstract

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.