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Nonlocal Schr\"odinger-Kirchhoff equations with external magnetic field
Analysis of PDEs · Submitted 18 May 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The paper deals with existence and multiplicity of solutions of the fractional Schr\"{o}dinger--Kirchhoff equation involving an external magnetic potential.
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The paper deals with existence and multiplicity of solutions of the fractional Schr\"{o}dinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2\theta-2})(-\Delta)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $\theta\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-\Delta )_A^s$ is the fractional magnetic operator. In the super- and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The paper deals with existence and multiplicity of solutions of the fractional Schr\"{o}dinger--Kirchhoff equation involving an external magnetic potential. | contribution 0.79 | 0.75 | 0.11 | asserted 0.89 |
| s1 | As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2\theta-2})(-\Delta)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $\theta\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-\Delta )_A^s$ is the fractional magnetic operator. | interpretation 0.51 | 0.66 | 0.17 | asserted 0.91 |
| s2 | In the super- and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. | result 0.54 | 0.93 | 0.31 | asserted 1.00 |
| s3 | In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem. | result 0.75 | 0.90 | 0.17 | asserted 0.92 |
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Original abstract
The paper deals with existence and multiplicity of solutions of the fractional Schr\"{o}dinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2\theta-2})(-\Delta)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $\theta\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-\Delta )_A^s$ is the fractional magnetic operator. In the super- and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.
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What it cites · 3 sentences without a confident label
Uses the method or tool of 1 paper
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Following Lemma [ref] and using the same discussion of [ref] , we have
Ground states for fractional magnetic operators · Functional setup · 0.89
Cites as background 2 papers
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For details on fractional magnetic operators we refer to [ref] and to the references [ref] for the physical background.
Ground states for fractional magnetic operators · Introduction and main result · 1.00
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Indeed, the literature on nonlocal fractional operators and on their applications is quite large, see for example the recent monograph [ref] , the extensive paper [ref] and the references cited there.
Fractional elliptic problems with critical growth in the whole of $\R^n$ · Introduction and main result · 1.00
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