Optimal Liouville theorem for a semilinear Ornstein-Uhlenbeck equation

TL;DR

This paper proves that under certain conditions, all bounded solutions of a semilinear Ornstein-Uhlenbeck equation are constant, extending Liouville-type theorems to specific parameter regimes and solution classes.

Contribution

It establishes new Liouville theorems for bounded solutions of the semilinear Ornstein-Uhlenbeck equation, including radial solutions in the critical case.

Findings

All bounded solutions are constant for Sobolev subcritical or critical p with λ ≤ 1.

Radial solutions are also constant in the critical case when n ≥ 4 and λ in [3n/(2(n-1)), 2].

The results extend classical Liouville theorems to this class of equations.

Abstract

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