Abstract
We consider a kind of nonlinear systems on a locally finite graph \(G=(V,E)\). We prove via the mountain pass theorem that this kind of systems has a nontrivial ground state solution which depends on the parameter \(\lambda \) with some suitable assumptions on the potentials. Moreover, we pay attention to the concentration behavior of these solutions and prove that as \(\lambda \rightarrow \infty \), these solutions converge to a ground state solution of a corresponding Dirichlet problem. Finally, we also provide some numerical experiments to illustrate our results.
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Xu, J., Zhao, L. Existence and Convergence of Solutions for Nonlinear Elliptic Systems on Graphs. Commun. Math. Stat. (2023). https://doi.org/10.1007/s40304-022-00318-2
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DOI: https://doi.org/10.1007/s40304-022-00318-2