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

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Abstract

We mainly investigated the continuous dependence on parameters of nontrivial solutions for a generalized poly-Laplacian system on the weighted finite graph G = (V,E). We first presented 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 generalization of those results in [6]. Then, we mainly showed 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 presented the existence result, the concrete upper and lower bounds, uniqueness, and dependence on parameters for the local minimum type nontrivial solutions. Subsequently, we presented an example on optimal control as an application of our results. Finally, we gave a nonexistence result and some results for the corresponding scalar equation.

Keywords

generalized ploy-Laplacian system / dependence on parameters / weighted finite graph / mountain pass type solution / local minimum type solution / optimal control

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