Symmetry of solutions to semilinear equationsinvolving the fractional laplacian

Lizhi  Zhang

doi:10.3934/cpaa.2015.14.2393

Article Contents

2015, Volume 14, Issue 6: 2393-2409. Doi: 10.3934/cpaa.2015.14.2393

This issue Previous Article The Liouville theorems for elliptic equations with nonstandard growth Next Article Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon

Symmetry of solutions to semilinear equations involving the fractional laplacian

Lizhi Zhang^1,

1.
School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007

Received: February 2015

Revised: July 2015

Published: November 2015

Abstract / Introduction Related Papers Cited by

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.

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Mathematics Subject Classification: Primary: 35S15, 35B06, 35J61.

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