Article Contents
Article Contents

# Symmetry of solutions to semilinear equations involving the fractional laplacian

• 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: $$\left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x)=f(x,u), & \qquad x\in\Omega, (1)\\ u(x)\equiv0, & \qquad x\notin\Omega. \end{array}\right.$$
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.
Mathematics Subject Classification: Primary: 35S15, 35B06, 35J61.

 Citation:

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