Liouville theorems for fractional  Hénon equation and system on $\mathbb{R}^n$

Dou, Jingbo; Zhou, Huaiyu

doi:10.3934/cpaa.2015.14.1915

Cited by 28 publications

(19 citation statements)

References 31 publications

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“…Moreover, Fall and Weth obtain some similar results in Theorem 1.2 in for some special indicators to fractional Laplacian operator

{(- normalΔ)}^{\frac{α}{2}}

. For more results about Liouville‐type theorems, see .…”

Section: Introductionsupporting

confidence: 52%

Positive solutions to semilinear elliptic equations involving a weighted fractional Lapalacian

Tang

2016

Math Methods in App Sciences

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In this paper, we consider a uniform elliptic nonlocal operator scriptAαu(x)=Cn,αP.V.∫Rna(x−y)(u(x)−u(y))|x−y|n+αdy, which is a weighted form of fractional Laplacian. We firstly establish three maximum principles for antisymmetric functions with respect to the nonlocal operator. Then, we obtain symmetry, monotonicity, and nonexistence of solutions to some semilinear equations involving the operator scriptAα on bounded domain, double-struckR+n and double-struckRn, by applying direct moving plane methods. Finally, we show the relations between the classical operator − Δ and the nonlocal operator in () as α→2. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Moreover, Fall and Weth obtain some similar results in Theorem 1.2 in for some special indicators to fractional Laplacian operator

{(- normalΔ)}^{\frac{α}{2}}

. For more results about Liouville‐type theorems, see .…”

Section: Introductionsupporting

confidence: 52%

Positive solutions to semilinear elliptic equations involving a weighted fractional Lapalacian

Tang

2016

Math Methods in App Sciences

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“…where 0 < α < 2, 1 < p, q ≤ n+α n-α , was obtained by Cai and Mei [39]. Using the method of moving plane in integral forms, Dou and Zhou in [40] proved the Liouville theorem of the positive solutions to the following fractional Henon system in R n :…”

Section: Introductionmentioning

confidence: 99%

Non-existence for a semi-linear fractional system with Sobolev exponents via direct method of moving spheres

Niu

2020

Bound Value Probl

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The aim of this article is to consider the semi-linear fractional system with Sobolev exponents q = n+α n-β and p = n+β n-α (α = β): (-) α/2 u(x) = k(x)v q (x) + f (v(x)), (-) β/2 v(x) = j(x)u p (x) + g(u(x)), where 0 < α, β < 2. We first establish two maximum principles for narrow regions in the ball and out of the ball by the iteration technique, respectively. Based on these principles, we use the direct method of moving spheres to prove the non-existence of positive solutions to the above system in the whole space and bounded star-shaped domain. As a consequence, the monotonic decreasing properties of W(x) = |x| n-α 2 u(x) and W 1 (x) = |x| n-β 2 v(x) along the radial direction in the whole space are obtained.

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“…The nonlinear terms in (1.10) is called critical if p = p s (a) := n+α+2a n−α (:= +∞ if n = α), subcritical if 0 < p < p s (a) and supercritical if p s (a) < p < +∞. Liouville type theorems for equations (1.10) (i.e., nonexistence of nontrivial nonnegative solutions) in the whole space R n , the half space R n + and bounded domains Ω have been extensively studied (see [1,2,3,4,5,7,10,13,15,16,17,18,19,20,21,23,28,29,33,36,37,38,39] and the references therein). For other related properties on PDEs (1.10) and Liouville type theorems on systems of PDEs of type (1.10) with respect to various types of solutions (e.g., stable, radial, singular, nonnegative, sign-changing, • • • ), please refer to [1,3,6,12,14,16,18,22,27,28,29,32,35,39] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains

Dai,

Qin

2019

Preprint

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In this paper, we are mainly concerned with the Dirichlet problems in exterior domains for the following elliptic equations:with arbitrary r > 0, where n ≥ 2, 0 < α ≤ 2 and f (x, u) satisfies some assumptions. A typical case is the Hardy-Hénon type equations in exterior domains. We first derive the equivalence between (0.1) and the corresponding integral equationswhere G α (x, y) denotes the Green's function for (−∆) α 2 in Ω r with Dirichlet boundary conditions. Then, we establish Liouville theorems for (0.2) via the method of scaling spheres developed in [17] by Dai and Qin, and hence obtain the Liouville theorems for (0.1). Liouville theorems for integral equations related to higher order Navier problems in Ω r are also derived.

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Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$

Cited by 28 publications

References 31 publications

Positive solutions to semilinear elliptic equations involving a weighted fractional Lapalacian

Positive solutions to semilinear elliptic equations involving a weighted fractional Lapalacian

Non-existence for a semi-linear fractional system with Sobolev exponents via direct method of moving spheres

Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains

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