Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
Wanting Qi, Xingyong Zhang · Demonstratio Mathematica · 2025
Abstract We investigate the existence and multiplicity of solutions for a class of the generalized coupled system involving poly-Laplacian and the parameter λ \lambda on finite graphs. By using the Mountain pass lemma together with the cut-off technique, we obtain that system has at least a nontrivial weak solution ( u λ , v λ ) \left({u}_{\lambda },{v}_{\lambda }) for every large parameter λ \lambda when the nonlinear term F ( x , u , v ) F\left(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 λ \lambda and the trend of ( u λ , v λ ) \left({u}_{\lambda },{v}_{\lambda }) with the change of λ \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 λ > 0 \lambda \gt 0 when the nonlinear term F ( x , u , v ) F\left(x,u,v) satisfies sublinear growth conditions only in a neighborhood of origin point (0, 0).