Liouville-type theorems and existence results for stable solutions to weighted Lane–Emden equations

2019 ◽  
Vol 150 (3) ◽  
pp. 1567-1579
Author(s):  
Alberto Farina ◽  
Shoichi Hasegawa
Keyword(s):  
Euclidean Space ◽  
Existence Result ◽  
Large Family ◽  
Existence Results ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Stable Solutions

AbstractWe devote this paper to proving non-existence and existence of stable solutions to weighted Lane-Emden equations on the Euclidean space ℝN, N ⩾ 2. We first prove some new Liouville-type theorems for stable solutions which recover and considerably improve upon the known results. In particular, our approach applies to various weighted equations, which naturally appear in many applications, but that are not covered by the existing literature. A typical example is provided by the well-know Matukuma's equation. We also prove an existence result for positive, bounded and stable solutions to a large family of weighted Lane–Emden equations, which indicates that our Liouville-type theorems are somehow sharp.

scholarly journals Non-Existence Results for Stable Solutions to Weighted Elliptic Systems Including Advection Terms

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 252
Author(s):  
Suleman Alfalqi
Keyword(s):  
Elliptic Systems ◽  
Existence Results ◽  
Liouville Type ◽  
Non Linear ◽  
Liouville Type Theorems ◽  
Stable Solutions ◽  
Bootstrap Iteration ◽  
Comparison Property

In this paper, we study a non-linear weighted Grushin system including advection terms. We prove some Liouville-type theorems for stable solutions of the system, based on the comparison property and the bootstrap iteration. Our results generalise and improve upon some previous works.

Liouville type theorems for stable solutions ofp-Laplace equation inRN

2017 ◽  
Vol 160 ◽  
pp. 44-52 ◽  
Author(s):  
Caisheng Chen ◽  
Hongxue Song ◽  
Hongwei Yang
Keyword(s):  
Laplace Equation ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Stable Solutions

scholarly journals Finite Morse index solutions of the Hénon Lane–Emden equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Abdellaziz Harrabi ◽  
Cherif Zaidi
Keyword(s):  
Morse Index ◽  
Monotonicity Formula ◽  
Liouville Type ◽  
Emden Equation ◽  
Type Identity ◽  
Liouville Type Theorems ◽  
Whole Space ◽  
Stable Solutions ◽  
Pohozaev Type Identity

Abstract In this paper, we are concerned with Liouville-type theorems of the Hénon Lane–Emden triharmonic equations in whole space. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the Pohozaev-type identity, monotonicity formula of solutions and a blowing down sequence.

scholarly journals Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Foued Mtiri
Keyword(s):  
Elliptic System ◽  
Stable Solution ◽  
Integral Estimate ◽  
Liouville Type ◽  
Grushin Operator ◽  
Degenerate Elliptic System ◽  
Liouville Type Theorems ◽  
Degenerate Elliptic ◽  
Stable Solutions ◽  
Alpha Beta

<p style='text-indent:20px;'>We examine the following degenerate elliptic system:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{s} u \! = \! v^p, \quad -\Delta_{s} v\! = \! u^\theta, \;\; u, v&gt;0 \;\;\mbox{in }\; \mathbb{R}^N = \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p, \theta \!&gt;\!0. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove that the system has no stable solution provided <inline-formula><tex-math id="M1">\begin{document}$ p, \theta &gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ N_s: = N_1+(1+s)N_2&lt; 2 + \alpha + \beta, $\end{document}</tex-math></inline-formula> where</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>This result is an extension of some results in [<xref ref-type="bibr" rid="b15">15</xref>]. In particular, we establish a new integral estimate for <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ v $\end{document}</tex-math></inline-formula> (see Proposition 1.1), which is crucial to deal with the case <inline-formula><tex-math id="M5">\begin{document}$ 0 &lt; p &lt; 1. $\end{document}</tex-math></inline-formula></p>

scholarly journals Liouville Type Theorems for Stable Solutions of Certain Elliptic Systems

2012 ◽  
Vol 12 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Mostafa Fazly
Keyword(s):  
Elliptic Systems ◽  
Bounded Domains ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Stable Solutions

AbstractWe establish Liouville type theorems for elliptic systems with various classes of nonlinearities on ℝis necessarily constant, whenever the dimension N < 8 + 3α +We also consider the case of bounded domains Ω ⊂ ℝ

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