Potential Analysis

, Volume 46, Issue 3, pp 569–588 | Cite as

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

Article

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:
$$\left \{\begin {array}{l} (-{\Delta })^{\alpha /2}u(x)=v^{q}(x),\ \ \ x\in \mathbb {R}^{n}_{+},\\ (-{\Delta })^{\alpha /2}v(x)=u^{p}(x),\ \ \ x\in \mathbb {R}^{n}_{+},\\ u(x)=v(x)=0,\ \ \ \ \ \ \ \ \ \ x\notin \mathbb {R}^{n}_{+}. \end {array}\right .\label {elliptic} $$
(1)
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

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

Mathematics Subject Classification (2010)

Primary: 35R11; Secondary: 35B53 45G15 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.School of Mathematics and Systems ScienceBeihang University (BUAA)BeijingPeople’s Republic of China
  2. 2.School of Mathematics and Computer ScienceJiangxi Science and Technology Normal UniversityNanchangPeople’s Republic of China
  3. 3.Department of MathematicsUniversity of ConnecticutStorrsUSA

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