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

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 $(0,0)$.