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A Liouville theorem for \(\alpha\)-harmonic functions in \(\mathbb{R}^n_+\). (English) Zbl 1338.35461

The purpose of the article is to show that a positive function \(u\) belonging to a suitable distribution space, and satisfying: \[ (*):\;(-\Delta)^su=0\text{ in }\mathbb R_+^n:=\{x=(x_l)_{1\leq l\leq n}\in \mathbb R^n \text{ s.t. } x_n>0\},\text{ and } u=0\text{ in } \mathbb R^n\setminus\mathbb R_+^n, \] is given explicitly, where \((-\Delta)^s\) is the classical fractional Laplacian operator with \(s\in (0,1)\). Precisely, the authors state that positive solutions associated to \((*)\) are of the form \(Cx_n^s\) (\(C\) is some positive constant), otherwise they are identically equal to zero (Theorem 1.1). The proof is essentially based on using Theorem 1.2 which states that a positive solution for \((*)\) has the following Poisson representation \[ u(x)=\int_{B_r^c(x_r)}P_r(x-x_r,y-x_r)u(y)dy, \] such that \(P_r(x-x_r,y-x_r)\) is the explicit Poisson kernel for \(B_r(x_r)\), the open ball centered at \(x_r=(0,\dots,r)\in\mathbb R_+^n\) with radius \(r\), and \(B^c_r(x_r):=\mathbb R^n\setminus B_r(x_r)\). Then the rest of the proof is based on resolving the systems \[ \left(\frac{\partial u(x)}{\partial x_l}=0\right)_{1\leq l\leq n-1} \] and \[ \frac{ x_n\partial u(x)}{\partial x_n}=su(x). \] These systems come from the derivatives of \(P_r(x-x_r,y-x_r)\) and then by taking \(r\) close to infinity.

MSC:

35R11 Fractional partial differential equations
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35B09 Positive solutions to PDEs
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