Liouville theorems involving the fractional Laplacian on a half space

Chen, Wenxiong; Fang, Yanqin; Yang, Ray

doi:10.1016/j.aim.2014.12.013

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Linear Equation1

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Cited by 168 publications

(90 citation statements)

References 18 publications

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“…By the Maximum principle [3,4,7] we have 0 ≤ u j (x) ≤ 1 on R n . This solution can be obtained by the variation method or the monotone method.…”

Section: Linear Equationmentioning

confidence: 99%

Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation

2016

Int. J. Math.

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“…By the Maximum principle [3,4,7] we have 0 ≤ u j (x) ≤ 1 on R n . This solution can be obtained by the variation method or the monotone method.…”

Section: Linear Equationmentioning

confidence: 99%

Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation

2016

Int. J. Math.

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“…Inspired by [17], in order to obtain the properties (symmetry and monotonicity) for the solutions of (1.10), we will use the method of moving planes in integral form for lack of a corresponding maximum principle for (1.10). For this, we need the Green's function…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, there are many literatures to study the properties of the solutions of this kind of equations. We can refer to [17][18][19][20][21][22][23][24] and the references therein. Chen et al [17] studied the following fractional Laplacian…”

Section: Introductionmentioning

confidence: 99%

The overdetermined equations on bounded domain

Yin

Zhang

Shang

2016

Complex Variables and Elliptic Equations

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In this paper we consider the equationon bounded domain ( ⊂ R n ) with Dirichlet conditions under two cases. One is 2 < α < n, the other is 0 < α < 2, where n ≥ 3. We will be concerned with the corresponding integral equations for lack of the maximum principle of the operator ( − ) α 2 in . We show the radial symmetry for and the symmetry and monotonicity for the positive solution of the equation through the method of moving planes in integral form which includes some innovative thoughts. ARTICLE HISTORY

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“…The direct method of moving planes is very useful, and a series of fruitful results have been obtained. For more articles concerning the method of moving planes for nonlocal equations and systems, mainly for integral equations, we refer to [13–21]. …”

Section: Introductionmentioning

confidence: 99%

Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian

Meng

Yang

2018

J Inequal Appl

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In this paper, we consider the following nonlinear Schrödinger system involving the fractional Laplacian operator: where . When Ω is the unit ball or , we prove that the solutions are radially symmetric and decreasing. When Ω is the parabolic domain on , we prove that the solutions are increasing. Furthermore, if Ω is the , then we also derive the nonexistence of positive solutions to the system on the half-space. We assume that the nonlinear terms f, g and the solutions u, v satisfy some amenable conditions in different cases.

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Liouville theorems involving the fractional Laplacian on a half space

Cited by 168 publications

References 18 publications

Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation

Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation

The overdetermined equations on bounded domain

Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian

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