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

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 (uλ,vλ)\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 (uλ,vλ)\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).

Keywords

• coupled system on finite graphs
• poly-laplacian
• local nonlinearity near origin
• mountain pass lemma
• clark’s theorem
• existence
• infinitely many nontrivial solutions
• 35j50
• 35j62
• 35b38
• 35r02