A new critical curve for the Lane-Emden system

Chen, Wenjing; Dupaigne, Louis; Ghergu, Marius

doi:10.3934/dcds.2014.34.2469

Cited by 11 publications

(12 citation statements)

References 20 publications

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“…when ρ ≡ 1, the stable solutions of the corresponding Lane-Emden equation and system, or the biharmonic equation (corresponding to p = 1) have been widely studied by many authors. See for instance [8,18,2,11,1,6] and the references there in.…”

Section: Introductionmentioning

confidence: 99%

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

Hajlaoui¹,

Harrabi²,

Mtiri³

2017

Discrete &Amp; Continuous Dynamical Systems - A

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We examine the general weighted Lane-Emden systemin R N for some α ≥ 0 and A > 0. We prove some Liouville type results for stable solution and improve the previous works [2,9,12]. In particular, we establish a new comparison property (see Proposition 1.1 below) which is crucial to handle the case 1 < p ≤ 4 3 . Our results can be applied also to the weighted Lane-Emden equation −∆u = ρ(x)u p in R N .

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Section: Introductionmentioning

confidence: 99%

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

Hajlaoui¹,

Harrabi²,

Mtiri³

2017

Discrete &Amp; Continuous Dynamical Systems - A

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“…One now has two options for the notion of the stability of (5). Either one views the equation as a scalar equation and uses the standard notion (8), when we do this we will say u is a stable solution of (5) or we view the solution as a solution of the system and we use the notion defined in (7), when we do this we will say (u, v) is a stable solution of (4) with p = 1. See Lemma 7 for a relationship between these notions of stability.…”

Section: Introductionmentioning

confidence: 99%

“…In [36] the range of exponents in Theorem 2 is improved. In [8] they examine (4) but without any stability assumptions. They obtain optimal results regarding the existence versus nonexistence of positive radial solutions of (4).…”

Section: Introductionmentioning

confidence: 99%

Liouville theorems for stable Lane–Emden systems and biharmonic problems

Cowan

2013

Nonlinearity

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We examine the elliptic system given byfor 1 < p ≤ θ and the fourth order scalar equationwhere 1 < θ. We prove various Liouville type theorems for positive stable solutions. For instance we show there are no positive stable solutions of (1) (resp. (2)) provided N ≤ 10 and 2 ≤ p ≤ θ (resp. N ≤ 10 and 1 < θ). Results for higher dimensions are also obtained. These results regarding stable solutions on the full space imply various Liouville theorems for positive (possibly unstable) bounded solutions ofwith u = v = 0 on ∂R N + . In particular there is no positive bounded solution of (3) for any 2 ≤ p ≤ θ if N ≤ 11. Higher dimensional results are also obtained.

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“…Note that it is not possible to use the classic Liouville results here, because we consider nonlinearities which are not necessarily asymptotic to a power at

+\infty

. Let us also note that there are many works where the classic Liouville results are used, see for example [1, 6, 7, 9, 16, 26].…”

Section: Introductionmentioning

confidence: 99%

Existence of solution of some p,q${p,q}$‐Laplacian system under local superlinear conditions

Cerda

Souto

Ubilla

2022

Mathematische Nachrichten

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We study existence of positive radial solutions for the following class of quasi‐linear elliptic systems 162.0pt{−normalΔpu=f(|x|,u,v)inB,−normalΔqv=g(|x|,u,v)inB,(u,v)=(0,0)on∂B,\begin{equation*}\hspace*{13.5pc} \left\lbrace \def\eqcellsep{&}\begin{array}{rcll} -\Delta _p u &=& f(|x|,u,v) & \text{in}\quad B, \\ -\Delta _q v &=& g(|x|,u,v) & \text{in}\quad B, \\ (u,v) &=& (0,0) & \text{on}\quad \partial B, \end{array} \right. \end{equation*}where the nonlinearities f,g∈Ctrue(B×false[0,+∞false)2;0.16em[0,+∞)true)$ f, g \in C\big (B \times [0,+\infty )^2;\,[0,+\infty )\big )$ satisfy some local superlinear property at +∞$+\infty$. Here B is the unity ball in double-struckRN${\mathbb {R}}^N$ and 1

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A new critical curve for the Lane-Emden system

Abstract: International audienceWe study stable positive radially symmetric solutions for the Lane- Emden system. We obtain a new critical curve that optimally describes the existence of such solutions

Cited by 11 publications

References 20 publications

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

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

Liouville theorems for stable Lane–Emden systems and biharmonic problems

Existence of solution of some p,q${p,q}$‐Laplacian system under local superlinear conditions

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