Existence and multiplicity of nontrivial solutions for poly-Laplacian systems on finite graphs

Abstract

Abstract In this paper, we investigate the existence and multiplicity of nontrivial solutions for poly-Laplacian system on a finite graph G = ( V , E ) $G=(V, E)$ , which is a generalization of the Yamabe equation on a finite graph. When the nonlinear term F satisfies the super- ( p , q ) $(p, q)$ -linear growth condition, by using the mountain pass theorem we obtain that the system has at least one nontrivial solution, and by using the symmetric mountain pass theorem, we obtain that the system has at least dim W nontrivial solutions, where W is the working space of the poly-Laplacian system. We also obtain the corresponding result for the poly-Laplacian equation. In some sense, our results improve some results in (Grigor’yan et al. in J. Differ. Equ. 261(9):4924–4943, 2016).

Keywords

• Mountain pass theorem
• Poly-Laplacian system
• Finite graph
• Super- ( p , q ) $(p, q)$ -linear growth condition