关键词: radial symmetry, fractional Laplacian, method of moving planes

Abstract: This paper deals with the radial symmetry of positive solutions to the nonlocal problem
$$(-\Delta)^s_\gamma u =b(x) f(u)\quad {\rm in}\ \, B_1\setminus\{0\},\qquad u=h\quad {\rm in}\ \, \mathbb{R}^N\setminus B_1, $$
where $b\!:B_1\to\mathbb{R}$ is locally Hölder continuous, radially symmetric and decreasing in the $|x|$ direction, $f\!: \mathbb{R}\to\mathbb{R}$ is a Lipschitz function, $h\!:B_1\to\mathbb{R}$ is radially symmetric, decreasing with respect to $|x|$ in $\mathbb{R}^N\setminus B_1$, $B_1$ is the unit ball centered at the origin, and $(-\Delta)^s_\gamma $ is the weighted fractional Laplacian with $s\in(0,1), \gamma\in[0,2s)$ defined by
$$(-\Delta)_{\gamma}^s u(x) =c_{N,s}\lim_{\delta \rightarrow 0^{+}}\int_{\mathbb{R}^N\backslash{B_{\delta}(x)}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}|y|^{\gamma}{\rm d}y.$$
We consider the radial symmetry of isolated singular positive solutions to the nonlocal problem in whole space
$$(-\Delta)^s_\gamma u(x)=b(x) f(u)\quad {\rm in}\ \, \mathbb{R}^N\setminus\{0\} ,$$
under suitable additional assumptions on $b$ and $f$. Our symmetry results are derived by the method of moving planes, where the main difficulty comes from the weighted fractional Laplacian. Our results could be applied to get a sharp asymptotic for semilinear problems with the fractional Hardy operators
$$(-\Delta)^s u +\frac{\mu}{|x|^{2s}}u=b(x) f(u)\quad {\rm in}\ \, B_1\setminus\{0\},\qquad u=h\quad {\rm in}\ \, \mathbb{R}^N\setminus B_1,$$
under suitable additional assumptions on $b$, $f$ and $h$.

Key words: radial symmetry, fractional Laplacian, method of moving planes