Optimal Liouville theorem for a semilinear Ornstein-Uhlenbeck equation

The question of triviality of solutions of the semilinear Ornstein-Uhlenbeck equation, $$\mathrm{\Delta }w-\frac{1}{2}⟨x,\mathrm{\nabla }w⟩-\frac{\lambda }{p-1}w+\mathrm{\mid }w{\mathrm{\mid }}^{p-1}w=0,$$ is considered. It is shown, that if $p>1$ is Sobolev subcritical or critical and $\lambda \le 1$, then all bounded entire solutions are constant. Moreover, in the critical case, the same conclusion holds in the subclass of radial solutions provided that $n\ge 4$ and $\lambda \in [\frac{3n}{2(n-1)},2]$.