Infinitely many solutions for two generalized poly-Laplacian systems on weighted graphs

Abstract

We investigate the multiplicity of solutions for a generalized poly-Laplacian system on weighted finite graphs and a generalized poly-Laplacian system with Dirichlet boundary value on weighted locally finite graphs, respectively, via the variational methods which are based on mountain pass theorem and topological degree theory. We obtain that these two systems have a sequence of minimax type solutions $\{(u_n,v_n)\}$ satisfying the energy functional $\ddot{I}†(u_n,v_n)\to +\infty$ as $n\to +\infty$ and a sequence of local minimum type solutions $\{(u_m^{\ast },v_m^{\ast })\}$ satisfying the energy functional $\ddot{I}†(u_m^{\ast },v_m^{\ast })\to -\infty$ as $m\to +\infty$.