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

Abstract

We investigate the existence and multiplicity of solutions for a class of generalized coupled system involving poly-Laplacian and a parameter $\hat{I}»$ on finite graphs. By using mountain pass lemma together with cut-off technique, we obtain that system has at least a nontrivial weak solution $(u_{\hat{I}}»,v_{\hat{I}}»)$ for every large parameter $\hat{I}»$ 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 $\hat{I}»$ and the trend of $(u_{\hat{I}}»,v_{\hat{I}}»)$ with the change of parameter $\hat{I}»$. 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 $\hat{I}»>0$ when the nonlinear term $F(x,u,v)$ satisfies sublinear growth conditions only in a neighborhood of origin point $(0,0)$.