Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space

Wei Dai; Zhao Liu; Guozhen Lu

doi:10.1007/s11118-016-9594-6

Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space

Wei Dai
, Zhao Liu
, Guozhen Lu^*

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

42 Scopus citations

Abstract

In this paper, let α be any real number between 0 and 2, we study the Dirichlet problem for semi-linear elliptic system involving the fractional Laplacian:(Formula presented.) We will first establish the equivalence between PDE problem (1) and the corresponding integral equation (IE) system (Lemma 2). Then we use the moving planes method in integral forms to establish our main theorem, a Liouville type theorem for the integral system (Theorem 3). Then we conclude the Liouville type theorem for the above differential system involving the fractional Laplacian (Corollary 4).

Original language	English
Pages (from-to)	569-588
Number of pages	20
Journal	Potential Analysis
Volume	46
Issue number	3
DOIs	https://doi.org/10.1007/s11118-016-9594-6
State	Published - 1 Mar 2017

Keywords

Dirichlet problem
Half space
Liouville type theorem
Method of moving planes in integral forms
Nonexistence
Rotational symmetry
The fractional Laplacian

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10.1007/s11118-016-9594-6

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title = "Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space",

abstract = "In this paper, let α be any real number between 0 and 2, we study the Dirichlet problem for semi-linear elliptic system involving the fractional Laplacian:(Formula presented.) We will first establish the equivalence between PDE problem (1) and the corresponding integral equation (IE) system (Lemma 2). Then we use the moving planes method in integral forms to establish our main theorem, a Liouville type theorem for the integral system (Theorem 3). Then we conclude the Liouville type theorem for the above differential system involving the fractional Laplacian (Corollary 4).",

keywords = "Dirichlet problem, Half space, Liouville type theorem, Method of moving planes in integral forms, Nonexistence, Rotational symmetry, The fractional Laplacian",

author = "Wei Dai and Zhao Liu and Guozhen Lu",

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year = "2017",

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doi = "10.1007/s11118-016-9594-6",

language = "英语",

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TY - JOUR

T1 - Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space

AU - Dai, Wei

AU - Liu, Zhao

AU - Lu, Guozhen

PY - 2017/3/1

Y1 - 2017/3/1

N2 - In this paper, let α be any real number between 0 and 2, we study the Dirichlet problem for semi-linear elliptic system involving the fractional Laplacian:(Formula presented.) We will first establish the equivalence between PDE problem (1) and the corresponding integral equation (IE) system (Lemma 2). Then we use the moving planes method in integral forms to establish our main theorem, a Liouville type theorem for the integral system (Theorem 3). Then we conclude the Liouville type theorem for the above differential system involving the fractional Laplacian (Corollary 4).

AB - In this paper, let α be any real number between 0 and 2, we study the Dirichlet problem for semi-linear elliptic system involving the fractional Laplacian:(Formula presented.) We will first establish the equivalence between PDE problem (1) and the corresponding integral equation (IE) system (Lemma 2). Then we use the moving planes method in integral forms to establish our main theorem, a Liouville type theorem for the integral system (Theorem 3). Then we conclude the Liouville type theorem for the above differential system involving the fractional Laplacian (Corollary 4).

KW - Dirichlet problem

KW - Half space

KW - Liouville type theorem

KW - Method of moving planes in integral forms

KW - Nonexistence

KW - Rotational symmetry

KW - The fractional Laplacian

UR - https://www.scopus.com/pages/publications/84988958345

U2 - 10.1007/s11118-016-9594-6

DO - 10.1007/s11118-016-9594-6

M3 - 文章

AN - SCOPUS:84988958345

SN - 0926-2601

VL - 46

SP - 569

EP - 588

JO - Potential Analysis

JF - Potential Analysis

IS - 3

ER -

Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space

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Keywords

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