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Existence and axial symmetry of minimal action odd solutions for 2-D Schr\"{o}dinger-Newton equation
Analysis of PDEs · Submitted 6 Dec 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Using variational method with the Cerami compactness property, we prove the existence of minimal action odd solutions.
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The abstract, sentence by sentence
We consider the following 2-D Schr\"{o}dinger-Newton equation \begin{eqnarray*} \begin{cases} -\Delta u+u=w|u|^{p-1}u \\ -\Delta w=2 \pi |u|^p \end{cases}\text{in} \; \mathbb{R}^2 \end{eqnarray*} for $ p \geq 2 $. Using variational method with the Cerami compactness property, we prove the existence of minimal action odd solutions. Also by carefully applying the method of moving plane to a similar but more complex equation on the upper half space, we prove these solutions are in fact axially symmetric. Our results partially can be seen as the counterpart of results in paper \cite{GS} for the 2-D case, or the extension of the results \cite{CW} to the odd solution case.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the following 2-D Schr\"{o}dinger-Newton equation \begin{eqnarray*} \begin{cases} -\Delta u+u=w|u|^{p-1}u \\ -\Delta w=2 \pi |u|^p \end{cases}\text{in} \; \mathbb{R}^2 \end{eqnarray*} for $ p \geq 2 $. | background 0.96 | 0.26 | 0.03 | unclear 0.56 |
| s1 | Using variational method with the Cerami compactness property, we prove the existence of minimal action odd solutions. | contribution 0.45 | 0.97 | 0.52 | asserted 1.00 |
| s2 | Also by carefully applying the method of moving plane to a similar but more complex equation on the upper half space, we prove these solutions are in fact axially symmetric. | result 0.80 | 0.95 | 0.26 | asserted 1.00 |
| s3 | Our results partially can be seen as the counterpart of results in paper \cite{GS} for the 2-D case, or the extension of the results \cite{CW} to the odd solution case. | interpretation 0.85 | 0.33 | 0.04 | hedged possibility 0.89 |
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Original abstract
We consider the following 2-D Schr\"{o}dinger-Newton equation \begin{eqnarray*} \begin{cases} -\Delta u+u=w|u|^{p-1}u \\ -\Delta w=2 \pi |u|^p \end{cases}\text{in} \; \mathbb{R}^2 \end{eqnarray*} for $ p \geq 2 $. Using variational method with the Cerami compactness property, we prove the existence of minimal action odd solutions. Also by carefully applying the method of moving plane to a similar but more complex equation on the upper half space, we prove these solutions are in fact axially symmetric. Our results partially can be seen as the counterpart of results in paper \cite{GS} for the 2-D case, or the extension of the results \cite{CW} to the odd solution case.
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