Symmetry of components and Liouville-type theorems for semilinear elliptic systems involving the fractional Laplacian

Yang, Hui; Zou, Wenming

doi:10.1016/j.na.2018.10.006

Cited by 13 publications

(9 citation statements)

References 39 publications

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“…and  3R (x) is the ball in R N+1 with radius 3R and its center at the x;  + 3R =  3R ∩ R N+1 + is the upper half ball; and ′  + 3R is the flat part of  + 3R , which is the ball B 3R in R N . For other results of fractional Laplacian equations, please see some works [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] and reference therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Harnack‐type inequality for fractional elliptic equations with critical exponent

Huang

Tian

2020

Math Methods in App Sciences

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Harnack‐type inequality for fractional elliptic equations with critical exponent

Huang

Tian

2020

Math Methods in App Sciences

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“…Inspired by [40,36,31], we establish the following a priori integral estimates for solutions to the fractional Lane-Emden system (1.1).…”

Section: Preliminariesmentioning

confidence: 99%

“…Proof. The proof adapt an idea of [40], originally coming from [34]. Let w := v − lu σ , where σ = q+1 p+1 and l = σ − 1 p+1 , the proof of (2.7) consists to show that (−∆) s w ≤ 0 in the set {w ≥ 0}.…”

Section: Preliminariesmentioning

confidence: 99%

Liouville type theorems for solutions of the weighted fractional Lane-Emden system

Hajlaoui¹

2022

Preprint

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“…Inspired by [40,36,31], we establish the following a priori integral estimates for solutions to the fractional Lane-Emden system (1). Lemma 2.1.…”

mentioning

confidence: 99%

“…Proof. The proof adapt an idea of [40], originally coming from [34]. Let w := v−lu σ , where σ = q+1 p+1 and l = σ − 1 p+1 , the proof of ( 21) consists to show that…”

mentioning

confidence: 99%

Liouville type theorems for stable solutions of the weighted fractional Lane-Emden system

Hajlaoui

2022

DCDS

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In this paper, we prove Liouville type theorems for stable solutions to the weighted fractional Lane-Emden system<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} (-\Delta)^s u = h(x)v^p,\quad (-\Delta)^s v = h(x)u^q, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N, \end{align*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ 1<q\leq p $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ h $\end{document}</tex-math></inline-formula> is a positive continuous function in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> satisfying <inline-formula><tex-math id="M4">\begin{document}$ {\liminf_{|x|\to \infty}}\frac{h(x)}{|x|^\ell} > 0 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ \ell > 0. $\end{document}</tex-math></inline-formula> Our results generalize the results established in [<xref ref-type="bibr" rid="b23">23</xref>] for the Laplacian case (correspond to <inline-formula><tex-math id="M6">\begin{document}$ s = 1 $\end{document}</tex-math></inline-formula>) and improve the previous work [<xref ref-type="bibr" rid="b12">12</xref>]. As a consequence, we prove classification result for stable solutions to the weighted fractional Lane-Emden equation <inline-formula><tex-math id="M7">\begin{document}$ (-\Delta)^s u = h(x)u^p $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>.

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Symmetry of components and Liouville-type theorems for semilinear elliptic systems involving the fractional Laplacian

Cited by 13 publications

References 39 publications

Harnack‐type inequality for fractional elliptic equations with critical exponent

Harnack‐type inequality for fractional elliptic equations with critical exponent

Liouville type theorems for solutions of the weighted fractional Lane-Emden system

Liouville type theorems for stable solutions of the weighted fractional Lane-Emden system

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