Alvino–Ferone–Trombetti conjecture on extremal functions for Hardy–Sobolev–Maz'ya inequalities

Let $x=(y,z)\mathbb{R}^k\times\mathbb{R}^{n-k}$, with the parameters $n$, $k$, and $p$ as in inequality (1.1). Define

$$u_0(x)=\nu_0(y,z)=C_{p,n,k}\left[(1+|y|)^2+|z|^2\right]^{-\frac{n-p}{2(p-1)}},$$

where

$$C_{p,n,k}=(k-1)^{\frac{n-p}{p(p-1)}}(n-p)^{\frac{n-p}{p}}(p-1)^{-\frac{n-p}{p}}.$$

Alvino–Ferone–Trombetti's conjecture. Every extremal function $u$ for inequality (1.1) is of the form

$$u(y,z)=\lambda^{\frac{n-p}{p}}u_0(\lambda y,\lambda z+z_0)$$

for some $\lambda>0$ and $z_0\in\mathbb{R}^{n-k}$. The cited work establishes this only in the case $n=3$, $k=2$, and $p=2$; the conjecture concerns the corresponding classification in the general parameter range of the inequality.