Abstract
Motivated by problems arising in nonlinear optics and Bose–Einstein condensates, we consider in \(\mathbb R^N\) (\(N \le 3\)) the following \(n \times n\) system of coupled Schrödinger equations
where \(\varepsilon >0\) is a parameter, \(\beta _{ij}\) are constants satisfying \(\beta _{ii} > 0\), and \(V_i\) are positive potentials that admit some common critical points \(a_1, \ldots ,a_k\) satisfying certain non-degenerate assumption. Then for any subsets \(J\subset \{1,2,\ldots ,k\}\), using a Lyapunov–Schmidt reduction method, we prove the existence of multi-bump bound solutions which as \(\varepsilon \rightarrow 0\) concentrate on \(\cup _{j\in J}a_j\).
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Acknowledgements
This paper was partially done while the second author was visiting the Department of Mathematics at Vanderbilt University, and the Department of Mathematics at College of Staten Island (CUNY) supported by the AMS Fan Fund China Exchange program (2015). The author is grateful to both departments for their hospitality during his stay. The first author thanks the support of the Center of PDE at the East China Normal University (Shanghai) where this work was finalized.
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This work was partially supported by a grant from the Simons Foundation (No. 210368 to Marcello Lucia). The second author was supported by National Science Foundation of China (No. 11571040).
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Lucia, M., Tang, Z. Multi-bump bound states for a Schrödinger system via Lyapunov–Schmidt Reduction. Nonlinear Differ. Equ. Appl. 24, 65 (2017). https://doi.org/10.1007/s00030-017-0489-z
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DOI: https://doi.org/10.1007/s00030-017-0489-z
Keywords
- Nonlinear Schrödinger systems
- Non-degenerate critical points
- Multi-bump bound states
- Variational methods