Symmetry and symmetry breaking for the fractional Caffarelli-Kohn-Nirenberg inequality

In this paper, we will consider the fractional Caffarelli-Kohn-Nirenberg inequality \begin{equation*} {\Lambda} \left(\int_{\r^n}\frac{|u(x)|^{p}}{|x|^{{\beta} {p}}}\,dx\right)^{\frac{2}{p}}\leq \int_{\r^n}\int_{\r^n}\frac{(u(x)-u(y))^2}{|x-y|^{n+2\gamma}|x|^{{\alpha}}|y|^{{\alpha}}}\,dy\,dx \end{equation*} where $\gamma\in(0,1)$, $n\geq 2$, and $\alpha,\beta\in\r$ satisfy \begin{equation*} \alpha\leq \beta\leq \alpha+\gamma, \ -2\gamma<\alpha<\frac{n-2\gamma}{2}, \end{equation*} and the exponent $p$ is chosen to be \begin{equation*} p=\frac{2n}{n-2\gamma+2(\beta-\alpha)}, \end{equation*} such that the inequality is invariant under scaling. We shall extend the results for the local case. We first study the existence and nonexistence of extremal solutions. Our next goal is to show some results for the symmetry and symmetry breaking region for the minimizers. In order to get these results we reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in cylindrical variables and we provide a non-local ODE to find the radially symmetric solutions. We also get non-degeneracy of critical points and uniqueness of minimizers in the radial symmetry class.