Symmetry of Solutions to Semilinear Equations Involving the Fractional Laplacian on $\mathbb{R}^n$ and $\mathbb{R}^n_+$

Lizhi Zhang; Yongzhong Wang

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arXiv · 1610.00122

Symmetry of Solutions to Semilinear Equations Involving the Fractional Laplacian on $\mathbb{R}^n$ and $\mathbb{R}^n_+$

Lizhi Zhang·Yongzhong Wang

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Abstract

Let $0<\alpha<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{\alpha/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Applying a direct method of moving planes for the fractional Laplacian, we prove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and $\mathbb{R}^n_+$ under mild conditions on $f$.

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Lizhi Zhang, Yongzhong Wang. 2016-10-01. Symmetry of Solutions to Semilinear Equations Involving the Fractional Laplacian on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. https://arxiv.org/abs/1610.00122

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