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
| [1] |
|
| [2] |
|
| [3] |
|
| [4] |
|
| [5] |
|
| [6] |
|
| [7] |
|
| [8] |
|
| [9] |
|
| [10] |
|
| [11] |
|
| [12] |
|
| [13] |
|
| [14] |
|
| [15] |
|
| [16] |
|
| [17] |
|
| [18] |
|
| [19] |
|
| [20] |
|
| [21] |
|
| [22] |
|
| [23] |
|
| [24] |
|
| [25] |
|
| [26] |
|
| [27] |
|
| [28] |
|
| [29] |
|
| [30] |
|
| [31] |
|
/
| 〈 |
|
〉 |