Existence and Convergence of Solutions for Nonlinear Elliptic Systems on Graphs
Jinyan Xu , Liang Zhao
Communications in Mathematics and Statistics ›› 2024, Vol. 12 ›› Issue (4) : 735 -754.
Existence and Convergence of Solutions for Nonlinear Elliptic Systems on Graphs
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.
Nonlinear elliptic system / Locally finite graph / Ground state solution
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