Complete classification conjecture for critical nonlocal quasi-linear equation

Let $N$ and $p$ satisfy $1<p<\frac{N}{2}$, and let $u$ be a nonnegative weak solution in $D^{1,p}(\mathbb{R}^{N})$ of the $D^{1,p}(\mathbb{R}^{N})$-critical nonlocal quasi-linear equation considered in the paper. Let $\widehat{U}(x)=\widehat{U}(r)$, where $r=|x|$, denote the positive radial solution normalized by $\widehat{U}(0)=1$, $\widehat{U}'(0)=0$, and $\widehat{U}'(r)<0$ for $r>0$, assumed to be unique.

Complete classification conjecture. Either $u\equiv 0$, or there exist $\lambda:=u(0)^{\frac{p}{N-p}}>0$ and $x_{0}\in\mathbb{R}^{N}$ such that

$$\nu(x)=\lambda^{\frac{N-p}{p}}\widehat{U}(\lambda(x-x_{0})).$$

Thus every positive weak solution is uniquely determined by the normalized radial solution up to scaling and translation. The conjecture would classify all nonnegative $D^{1,p}(\mathbb{R}^{N})$-weak solutions, including minimizers of the associated minimization problem and extremal functions for the Hardy–Littlewood–Sobolev inequality; the uniqueness of the normalized positive radial solution and the resulting classification remain open in the source.