On the Stein-Weiss inequalities and higher-order Caffarelli-Kohn-Nirenberg type inequalities: sharp constants, symmetry of extremal functions

In this paper, we first classify all radially symmetry solutions of the following weighted fourth-order equation \begin{equation*} Δ(|x|^{-γ}Δu)=|x|^γu^{\frac{N+4+3γ}{N-4-γ}},\quad u\geq 0 \quad \mbox{in}\quad $\mathbb{R}$^N, \end{equation*} where $N\ge 5$, $-2<\gamma <0$. Then we derive the sharp Stein-Weiss inequality and standard second-order Caffarelli-Kohn-Nirenberg inequality with radially symmetry extremal functions. Moreover, by using standard spherical decomposition, we derive a sharp weighted Rellich-Sobolev inequality. Furthermore, we establish the sharp second-order Caffarelli-Kohn-Nirenberg type inequalities with two variables which have radially symmetry extremal functions. Finally, we derive the weak form Hardy-Rellich inequalities with sharp constants.