# Liouville theorem for fractional Laplacian

Is there any Liouville type theorem for the half space problem \begin{equation} \ \ \left\{\begin{aligned} (-\Delta)^s v &= f(v) &&\text{in } \mathbb R^N_+\\ v & =0 &&\text{in } \mathbb R^N \setminus \mathbb R^N_+\end{aligned} \right. \end{equation} where $$s\in (0, 1)$$ and $$v$$ changes sign. If $$v$$ is bounded and $$f(v)=0$$, does it imply $$v \equiv 0.$$

• I find it confusing if someone changes their question in a way which invalidates a previously correct answer. Did you mean to ask a new question? Jul 27, 2021 at 16:14

Yes, $$v$$ is necessarily zero.
The function $$v$$ in your question is $$\alpha$$-harmonic in the half-space $$H = \mathbb{R}^N_+$$. Lemma 17 in [K. Bogdan, The boundary Harnack principle for the fractional Laplacian. Studia Math. 123(1) (1997): 43–80] (link) asserts that if $$v$$ is an $$\alpha$$-harmonic function in an open bounded set $$D$$, then $$v$$ is regular $$\alpha$$-harmonic in $$D$$, that is, it is given by a Poisson-type integral. Therefore, if $$v$$ is additionally zero outside $$D$$, then it is zero everywhere.
In our case, $$H$$ is not bounded. However, one can apply the same proof as in Bogdan's paper: boundedness of $$D$$ is required only to assert that the exit time from $$D$$ is finite with probability $$1$$, and the exit time from $$H$$ has this property.