Symmetry of solutions to semilinear equations involving the fractional laplacian
Lizhi Zhang
Communications on Pure & Applied Analysis 2015, 14(6): 2393-2409 doi: 10.3934/cpaa.2015.14.2393
Let $0<\alpha<2$ be any real number. Let $\Omega\subset\mathbb{R}^n$ be a bounded or an unbounded domain which is convex and symmetric in $x_1$ direction. We investigate the following Dirichlet problem involving the fractional Laplacian: \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x)=f(x,u), & \qquad x\in\Omega,                                              (1)\\ u(x)\equiv0, & \qquad x\notin\Omega. \end{array}\right. \end{equation}
    Applying a direct method of moving planes for the fractional Laplacian, with the help of several maximum principles for anti-symmetric functions, we prove the monotonicity and symmetry of positive solutions in $x_1$ direction as well as nonexistence of positive solutions under various conditions on $f$ and on the solutions $u$. We also extend the results to some more complicated cases.
keywords: Dirichlet problem semi-linear elliptic equation maximum principle for anti-symmetric functions nonexistence of positive solutions narrow region principle a direct method of moving planes the fractional Laplacian decay at infinity. symmetry Monotonicity
A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$
Lizhi Zhang Congming Li Wenxiong Chen Tingzhi Cheng
Discrete & Continuous Dynamical Systems - A 2016, 36(3): 1721-1736 doi: 10.3934/dcds.2016.36.1721
In this paper, we consider $\alpha$-harmonic functions in the half space $\mathbb{R}^n_+$: \begin{equation} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2} u(x)=0,~u(x)\geq0, & \qquad x\in\mathbb{R}^n_+, \\ u(x)\equiv0, & \qquad x\notin\mathbb{R}^{n}_{+}. \end{array}\right.                      (1) \end{equation} We prove that all solutions of (1) are either identically zero or assuming the form \begin{equation} u(x)=\left\{\begin{array}{ll}Cx_n^{\alpha/2}, & \qquad x\in\mathbb{R}^n_+, \\ 0, & \qquad x\notin\mathbb{R}^{n}_{+}, \end{array}\right. \label{2} \end{equation} for some positive constant $C$.
keywords: Poisson representation. $\alpha$-harmonic functions The fractional Laplacian uniqueness of solutions Liouville theorem

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