Communications on Pure and Applied Analysis (CPAA)

Symmetry of solutions to semilinear equations involving the fractional laplacian

Pages: 2393 - 2409, Volume 14, Issue 6, November 2015      doi:10.3934/cpaa.2015.14.2393

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Lizhi Zhang - School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China (email)

Abstract: 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:  Monotonicity, symmetry, nonexistence of positive solutions, the fractional Laplacian, Dirichlet problem, semi-linear elliptic equation, a direct method of moving planes, maximum principle for anti-symmetric functions, narrow region principle, decay at infinity.
Mathematics Subject Classification:  Primary: 35S15, 35B06, 35J61.

Received: February 2015;      Revised: July 2015;      Available Online: September 2015.