18-09-2020
323

## 几何分析研讨班第三期-Symmetry and symmetry breaking for the fractional Caffarelli-Kohn-Nirenberg inequality-敖微微教授

\label{ineq_u}

{\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

for $\gamma\in(0,1)$, $n>2\gamma$, and $\alpha,\beta\in\r$ satisfy

\begin{equation*}\label{parameter}

\alpha\leq \beta\leq \alpha+\gamma, \ -2\gamma<\alpha<\frac{n-2\gamma}{2}

\end{equation*}

and

$$p=\frac{2n}{n-2\gamma+2(\beta-\alpha)}.$$

We first study the existence and nonexistence of extremal solutions to (\ref{ineq_u}). Our next goal is to show some results for the symmetry and symmetry breaking region for the minimizers. In order to get these result, we reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in cylindrical variables and we provide a non-local ODE to find the radially symmetric extremals. We also get the non-degeneracy and uniqueness of minimizers in the radial symmetry class. This is joint work with Azahara DelaTorre and Maria del Mar Gonzalez.

E-mail：mathruc@ruc.edu.cn