Dependence on parameters of solutions for a generalized poly-Laplacian system on weighted graphs

TL;DR

This paper studies how solutions to a generalized poly-Laplacian system on weighted graphs depend on parameters, establishing existence, bounds, and continuous dependence results, with applications to optimal control and scalar equations.

Contribution

It provides new existence and dependence results for solutions of a generalized poly-Laplacian system on weighted graphs, including bounds and uniqueness, extending previous work.

Findings

Existence of mountain pass solutions under super-$(p, q)$ growth.

Solutions are uniformly bounded and depend continuously on parameters.

Application to optimal control and scalar equations included.

Abstract

We mainly investigate the continuous dependence on parameters of nontrivial solutions for a generalized poly-Laplacian system on the weighted finite graph $G=(V, E)$. We firstly present an existence result of mountain pass type nontrivial solutions when the nonlinear term $F$ satisfies the super-$(p, q)$ linear growth condition which is a simple generalization of those results in [28]. Then we mainly show that the mountain pass type nontrivial solutions of the poly-Laplacian system are uniformly bounded for parameters and the concrete upper and lower bounds are given, and are continuously dependent on parameters. Similarly, we also present the existence result, the concrete upper and lower bounds, uniqueness, and dependence on parameters for the locally minimum type nontrivial solutions. Subsequently, we present an example on optimal control as an application of our results. Finally, we give a nonexistence result and some results for the corresponding scalar equation.