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 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_, v_) for every large parameter 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 and the trend of (u_, v_) with the change of parameter. 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 >0 when the nonlinear term F (x, u, v) satisfies sublinear growth conditions only in a neighborhood of origin point (0, 0).