Wang's classification conjecture for semilinear subelliptic equations on CR manifolds

Let $M$ be a closed, oriented, strictly pseudoconvex CR manifold of hypersurface type with pseudohermitian structure $\theta$, and suppose that

$$\operatorname{Ric}(Z,Z)\geqslant (n+1)\langle Z,Z\rangle_{L_\theta},\qquad \operatorname{Tor}(Z,Z)=0,$$

for every $Z\in T^{(1,0)}M$. Let $\alpha>1$, $\lambda>0$, and let $u\in C^\infty(M)$ be positive and satisfy

$$\Delta u-\lambda u+u^\alpha=0.$$

Wang's conjecture. If $1<\alpha\leqslant\frac{n+2}{n}$ and $\lambda\leqslant\frac{n}{2(\alpha-1)}$, then the only positive solution is $u\equiv\lambda^{\frac{1}{\alpha-1}}$. Otherwise, $\alpha=\frac{n+2}{n}$, $\lambda=\frac{n^2}{4}$, and $(M^{2n+1},\theta)$ is the standard CR sphere $(\mathbb S^{2n+1},\theta_c)$, with some $s\geqslant0$ and $\xi\in\mathbb S^{2n+1}$ such that

$$u(z)=c_{n,s}|\cosh s+(\sinh s)\langle z,\xi\rangle|^{-n},\qquad z\in\mathbb S^{2n+1}.$$

This conjecture extends the known classification at the critical exponent and is intended to classify all positive smooth solutions in the stated subcritical and critical parameter ranges under the curvature and torsion condition. Its resolution is not supplied in the source.