Volume 37, Issue 2
Classification of Positive Ground State Solutions with Different Morse Indices for Nonlinear $N$-Coupled Schrödinger System

Juncheng Wei & Maoding Zhen

Anal. Theory Appl., 37 (2021), pp. 230-266.

Published online: 2021-04

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  • Abstract

In this paper, we study the following $N$-coupled nonlinear Schrödinger system

c.JPG

where $n\leq3,$ $ N\geq3$, $\mu_{j}>0, $ $\beta_{i, j}=\beta_{j, i}>0$ are constants and $\beta_{j, j}=\mu_{j},$ $ j=1, \cdots, N.$ There have been intensive studies for the system on existence/non-existence and classification of ground state solutions when $N=2$. However, fewer results about the classification of ground state solution are available for $N\geq3$. In this paper, we first give a complete classification result on ground state solutions with Morse indices $1,$ $2$ or $3$ for three-coupled Schrödinger system. Then we generalize our results to $N$-coupled Schrödinger system for ground state solutions with Morse indices 1 and $N$. We show that any positive ground state solutions with Morse index 1 or Morse index $N$ must be the form of $(d_{1}w,d_{2}w,\cdots,d_{N}w)$ under suitable conditions, where $w$ is the unique positive ground state solution of certain equation. Finally, we generalize our results to fractional $N$-coupled Schrödinger system.

  • Keywords

Nonlinear Schrödinger system, unique ground state solution, variational method, Morse indices.

  • AMS Subject Headings

35B09, 35J47, 35J50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-37-230, author = {Juncheng and Wei and and 14897 and and Juncheng Wei and Maoding and Zhen and and 14898 and and Maoding Zhen}, title = {Classification of Positive Ground State Solutions with Different Morse Indices for Nonlinear $N$-Coupled Schrödinger System}, journal = {Analysis in Theory and Applications}, year = {2021}, volume = {37}, number = {2}, pages = {230--266}, abstract = {

In this paper, we study the following $N$-coupled nonlinear Schrödinger system

c.JPG

where $n\leq3,$ $ N\geq3$, $\mu_{j}>0, $ $\beta_{i, j}=\beta_{j, i}>0$ are constants and $\beta_{j, j}=\mu_{j},$ $ j=1, \cdots, N.$ There have been intensive studies for the system on existence/non-existence and classification of ground state solutions when $N=2$. However, fewer results about the classification of ground state solution are available for $N\geq3$. In this paper, we first give a complete classification result on ground state solutions with Morse indices $1,$ $2$ or $3$ for three-coupled Schrödinger system. Then we generalize our results to $N$-coupled Schrödinger system for ground state solutions with Morse indices 1 and $N$. We show that any positive ground state solutions with Morse index 1 or Morse index $N$ must be the form of $(d_{1}w,d_{2}w,\cdots,d_{N}w)$ under suitable conditions, where $w$ is the unique positive ground state solution of certain equation. Finally, we generalize our results to fractional $N$-coupled Schrödinger system.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2021.pr80.07}, url = {http://global-sci.org/intro/article_detail/ata/18773.html} }
TY - JOUR T1 - Classification of Positive Ground State Solutions with Different Morse Indices for Nonlinear $N$-Coupled Schrödinger System AU - Wei , Juncheng AU - Zhen , Maoding JO - Analysis in Theory and Applications VL - 2 SP - 230 EP - 266 PY - 2021 DA - 2021/04 SN - 37 DO - http://doi.org/10.4208/ata.2021.pr80.07 UR - https://global-sci.org/intro/article_detail/ata/18773.html KW - Nonlinear Schrödinger system, unique ground state solution, variational method, Morse indices. AB -

In this paper, we study the following $N$-coupled nonlinear Schrödinger system

c.JPG

where $n\leq3,$ $ N\geq3$, $\mu_{j}>0, $ $\beta_{i, j}=\beta_{j, i}>0$ are constants and $\beta_{j, j}=\mu_{j},$ $ j=1, \cdots, N.$ There have been intensive studies for the system on existence/non-existence and classification of ground state solutions when $N=2$. However, fewer results about the classification of ground state solution are available for $N\geq3$. In this paper, we first give a complete classification result on ground state solutions with Morse indices $1,$ $2$ or $3$ for three-coupled Schrödinger system. Then we generalize our results to $N$-coupled Schrödinger system for ground state solutions with Morse indices 1 and $N$. We show that any positive ground state solutions with Morse index 1 or Morse index $N$ must be the form of $(d_{1}w,d_{2}w,\cdots,d_{N}w)$ under suitable conditions, where $w$ is the unique positive ground state solution of certain equation. Finally, we generalize our results to fractional $N$-coupled Schrödinger system.

Juncheng Wei & Maoding Zhen. (1970). Classification of Positive Ground State Solutions with Different Morse Indices for Nonlinear $N$-Coupled Schrödinger System. Analysis in Theory and Applications. 37 (2). 230-266. doi:10.4208/ata.2021.pr80.07
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