Nontrivial solutions for a generalized poly-Laplacian system on finite graphs

TL;DR

This paper studies the existence and multiplicity of solutions for a generalized poly-Laplacian system on finite graphs, using variational methods to establish conditions for solutions under different growth assumptions.

Contribution

It introduces new results on solution existence and multiplicity for a poly-Laplacian system on finite graphs, employing mountain pass and Clark's theorems with cut-off techniques.

Findings

Existence of at least one nontrivial solution for large parameter λ under superlinear growth.

Existence of a sequence of solutions tending to zero under sublinear growth.

Explicit lower bounds for the parameter λ and solution behavior trends.

Abstract

We investigate the existence and multiplicity of solutions for a class of generalized coupled system involving poly-Laplacian and a parameter $\lambda$ on finite graphs. By using mountain pass lemma together with cut-off technique, we obtain that system has at least a nontrivial weak solution $(u_{\lambda},v_{\lambda})$ for every large parameter $\lambda$ when the nonlinear term $F(x,u,v)$ satisfies superlinear growth conditions only in a neighborhood of origin point $(0,0)$. We also obtain a concrete form for the lower bound of parameter $\lambda$ and the trend of $(u_{\lambda},v_{\lambda})$ with the change of parameter $\lambda$. Moreover, by using a revised Clark's theorem together with cut-off technique, we obtain that system has a sequence of solutions tending to 0 for every $\lambda>0$ when the nonlinear term $F(x,u,v)$ satisfies sublinear growth conditions only in a neighborhood of origin point $(0,0)$.