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We extend a result of Lieb [‘On the lowest eigenvalue of the Laplacian for the intersection of two domains’, Invent. Math. 74(3) (1983), 441–448] to the fractional setting. We prove that if A and B are two bounded domains in 
$\mathbb R^N$ and 
$\lambda _s(A)$, 
$\lambda _s(B)$ are the lowest eigenvalues of 
$(-\Delta )^s$, 
$0<s<1$, with Dirichlet boundary conditions, there exists some translation 
$B_x$ of B such that 
$\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)$. Moreover, without the boundedness assumption on A and B, we show that for any 
$\varepsilon>0$, there exists some translation 
$B_x$ of B such that 
$\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)+\varepsilon .$
We study the fractional parabolic Lichnerowicz equation $$ \begin{align*} v_t+(-\Delta)^s v=v^{-p-2}-v^p \quad\mbox{in } \mathbb R^N\times\mathbb R \end{align*} $$where 
$p>0$ and 
$ 0<s<1 $. We establish a Liouville-type theorem for positive solutions in the case 
$p>1$ and give a uniform lower bound of positive solutions when 
$0<p\leq 1$. In particular, when v is independent of the time variable, we obtain a similar result for the fractional elliptic Lichnerowicz equation $$ \begin{align*} (-\Delta)^s u=u^{-p-2}-u^p \quad\mbox{in }\mathbb R^N \end{align*} $$with 
$p>0$ and 
$0<s<1$. This extends the result of Brézis [‘Comments on two notes by L. Ma and X. Xu’, C. R. Math. Acad. Sci. Paris 349(5–6) (2011), 269–271] to the fractional Laplacian.
